Abelian Sandpile Model¶

Author: David Perkinson, Reed College

Introduction¶

These notes provide an introduction to Dhar’s abelian sandpile model (ASM) and to Sage Sandpiles, a collection of tools in Sage for doing sandpile calculations. For a more thorough introduction to the theory of the ASM, the papers Chip-Firing and Rotor-Routing on Directed Graphs [H] by Holroyd et al., and Riemann-Roch and Abel-Jacobi Theory on a Finite Graph by Baker and Norine [BN] are recommended. See also [PPW2013].

To describe the ASM, we start with a sandpile graph: a directed multigraph $$\Gamma$$ with a vertex $$s$$ that is accessible from every vertex. By multigraph, we mean that each edge of $$\Gamma$$ is assigned a nonnegative integer weight. To say $$s$$ is accessible from some vertex $$v$$ means that there is a sequence (possibly empty) of directed edges starting at $$v$$ and ending at $$s$$. We call $$s$$ the sink of the sandpile graph, even though it might have outgoing edges, for reasons that will be made clear in a moment.

We denote the vertex set of $$\Gamma$$ by $$V$$, and define $$\tilde{V} = V\setminus\{s\}$$. For any vertex $$v$$, we let $$\mbox{out-degree}(v)$$ (the out-degree of $$v$$) be the sum of the weights of all edges leaving $$v$$.

Configurations and divisors¶

A configuration on $$\Gamma$$ is an element of $$\ZZ_{\geq 0} \tilde{V}$$, i.e., the assignment of a nonnegative integer to each nonsink vertex. We think of each integer as a number of grains of sand being placed at the corresponding vertex. A divisor on $$\Gamma$$ is an element of $$\ZZ V$$, i.e., an element in the free abelian group on all of the vertices. In the context of divisors, it is sometimes useful to think of a divisor on $$\Gamma$$ as assigning dollars to each vertex, with negative integers signifying a debt.

Stabilization¶

A configuration $$c$$ is stable at a vertex $$v\in\tilde{V}$$ if $$c(v) < \mbox{out-degree}(v)$$. Otherwise, $$c$$ is unstable at $$v$$. A configuration $$c$$ (in itself) is stable if it is stable at each nonsink vertex. Otherwise, $$c$$ is unstable. If $$c$$ is unstable at $$v$$, the vertex $$v$$ can be fired (toppled) by removing $$\mbox{out-degree}(v)$$ grains of sand from $$v$$ and adding grains of sand to the neighbors of $$v$$, determined by the weights of the edges leaving $$v$$ (each vertex $$w$$ gets as many grains as the weight of the edge $$(v, w)$$ is, if there is such an edge). Note that grains that are added to the sink $$s$$ are not counted (i.e., they “disappear”).

Despite our best intentions, we sometimes consider firing a stable vertex, resulting in a “configuration” with a negative amount of sand at that vertex. We may also reverse-fire a vertex, absorbing sand from the vertex’s neighbors (including the sink).

Example. Consider the graph:

$$\Gamma$$

All edges have weight $$1$$ except for the edge from vertex 1 to vertex 3, which has weight $$2$$. If we let $$c = (5,0,1)$$ with the indicated number of grains of sand on vertices 1, 2, and 3, respectively, then only vertex 1, whose out-degree is 4, is unstable. Firing vertex 1 gives a new configuration $$c' = (1,1,3)$$. Here, $$4$$ grains have left vertex 1. One of these has gone to the sink vertex (and forgotten), one has gone to vertex 2, and two have gone to vertex 3, since the edge from 1 to 3 has weight $$2$$. Vertex 3 in the new configuration is now unstable. The Sage code for this example follows.

sage: g = {0:{},
....:      1:{0:1, 2:1, 3:2},
....:      2:{1:1, 3:1},
....:      3:{1:1, 2:1}}
sage: S = Sandpile(g, 0)   # create the sandpile
sage: S.show(edge_labels=true)  # display the graph


Create the configuration:

sage: c = SandpileConfig(S, {1:5, 2:0, 3:1})
sage: S.out_degree()
{0: 0, 1: 4, 2: 2, 3: 2}


Fire vertex $$1$$:

sage: c.fire_vertex(1)
{1: 1, 2: 1, 3: 3}


The configuration is unchanged:

sage: c
{1: 5, 2: 0, 3: 1}


Repeatedly fire vertices until the configuration becomes stable:

sage: c.stabilize()
{1: 2, 2: 1, 3: 1}


Alternatives:

sage: ~c             # shorthand for c.stabilize()
{1: 2, 2: 1, 3: 1}
sage: c.stabilize(with_firing_vector=true)
[{1: 2, 2: 1, 3: 1}, {1: 2, 2: 2, 3: 3}]


Since vertex 3 has become unstable after firing vertex 1, it can be fired, which causes vertex 2 to become unstable, etc. Repeated firings eventually lead to a stable configuration. The last line of the Sage code, above, is a list, the first element of which is the resulting stable configuration, $$(2,1,1)$$. The second component records how many times each vertex fired in the stabilization.

Since the sink is accessible from each nonsink vertex and never fires, every configuration will stabilize after a finite number of vertex-firings. It is not obvious, but the resulting stabilization is independent of the order in which unstable vertices are fired. Thus, each configuration stabilizes to a unique stable configuration.

Laplacian¶

Fix a total order on the vertices of $$\Gamma$$, thus leading to a labelling of the vertices by the numbers $$1, 2, \ldots, n$$ for some $$n$$ (in the given order). The Laplacian of $$\Gamma$$ is

$L := D-A$

where $$D$$ is the diagonal matrix of out-degrees of the vertices (i.e., the diagonal $$n \times n$$-matrix whose $$(i, i)$$-th entry is the out-degree of the vertex $$i$$) and $$A$$ is the adjacency matrix whose $$(i,j)$$-th entry is the weight of the edge from vertex $$i$$ to vertex $$j$$, which we take to be $$0$$ if there is no edge. The reduced Laplacian, $$\tilde{L}$$, is the submatrix of the Laplacian formed by removing the row and column corresponding to the sink vertex. Firing a vertex of a configuration is the same as subtracting the corresponding row of the reduced Laplacian.

Example. (Continued.)

sage: S.vertices()  # the ordering of the vertices
[0, 1, 2, 3]
sage: S.laplacian()
[ 0  0  0  0]
[-1  4 -1 -2]
[ 0 -1  2 -1]
[ 0 -1 -1  2]
sage: S.reduced_laplacian()
[ 4 -1 -2]
[-1  2 -1]
[-1 -1  2]


The configuration we considered previously:

sage: c = SandpileConfig(S, [5,0,1])
sage: c
{1: 5, 2: 0, 3: 1}


Firing vertex 1 is the same as subtracting the corresponding row of the reduced Laplacian from the configuration (regarded as a vector):

sage: c.fire_vertex(1).values()
[1, 1, 3]
sage: S.reduced_laplacian()[0]
(4, -1, -2)
sage: vector([5,0,1]) - vector([4,-1,-2])
(1, 1, 3)


Recurrent elements¶

Imagine an experiment in which grains of sand are dropped one-at-a-time onto a graph, pausing to allow the configuration to stabilize between drops. Some configurations will only be seen once in this process. For example, for most graphs, once sand is dropped on the graph, no sequence of additions of sand and stabilizations will result in a graph empty of sand. Other configurations—the so-called recurrent configurations—will be seen infinitely often as the process is repeated indefinitely.

To be precise, a configuration $$c$$ is recurrent if (i) it is stable, and (ii) given any configuration $$a$$, there is a configuration $$b$$ such that $$c = \mbox{stab}(a+b)$$, the stabilization of $$a + b$$.

The maximal-stable configuration, denoted $$c_{\mathrm{max}}$$, is defined by $$c_{\mathrm{max}}(v)=\mbox{out-degree}(v)-1$$ for all nonsink vertices $$v$$. It is clear that $$c_{\mathrm{max}}$$ is recurrent. Further, it is not hard to see that a configuration is recurrent if and only if it has the form $$\mbox{stab}(a+c_{\mathrm{max}})$$ for some configuration $$a$$.

Example. (Continued.)

sage: S.recurrents(verbose=false)
[[3, 1, 1], [2, 1, 1], [3, 1, 0]]
sage: c = SandpileConfig(S, [2,1,1])
sage: c
{1: 2, 2: 1, 3: 1}
sage: c.is_recurrent()
True
sage: S.max_stable()
{1: 3, 2: 1, 3: 1}


Adding any configuration to the max-stable configuration and stabilizing yields a recurrent configuration:

sage: x = SandpileConfig(S, [1,0,0])
sage: x + S.max_stable()
{1: 4, 2: 1, 3: 1}


Use & to add and stabilize:

sage: c = x & S.max_stable()
sage: c
{1: 3, 2: 1, 3: 0}
sage: c.is_recurrent()
True


Note the various ways of performing addition and stabilization:

sage: m = S.max_stable()
sage: (x + m).stabilize() == ~(x + m)
True
sage: (x + m).stabilize() == x & m
True


Burning Configuration¶

A burning configuration is a $$\ZZ_{\geq 0}$$-linear combination $$f$$ of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex is accessible from some vertex in the support of $$f$$ (in other words, for each vertex $$w$$, there is a path from some $$v \in \operatorname{supp} f$$ to $$w$$). The corresponding burning script gives the coefficients in the $$\ZZ_{\geq 0}$$-linear combination needed to obtain the burning configuration. So if $$b$$ is a burning configuration, $$\sigma$$ is its script, and $$\tilde{L}$$ is the reduced Laplacian, then $$\sigma\,\tilde{L} = b$$. The minimal burning configuration is the one with the minimal script (each of its components is no larger than the corresponding component of any other script for a burning configuration).

Given a burning configuration $$b$$ having script $$\sigma$$, and any configuration $$c$$ on the same graph, the following are equivalent:

• $$c$$ is recurrent;
• $$c + b$$ stabilizes to $$c$$;
• the firing vector for the stabilization of $$c + b$$ is $$\sigma$$.

The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.

Example.

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1},
....:      3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: G = Sandpile(g,0)
sage: G.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
sage: G.burning_config().values()
[2, 0, 1, 1, 0]
sage: G.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
sage: G.burning_script().values()
[1, 3, 5, 1, 4]
sage: matrix(G.burning_script().values())*G.reduced_laplacian()
[2 0 1 1 0]


Sandpile group¶

The collection of stable configurations forms a commutative monoid with addition defined as ordinary addition followed by stabilization. The identity element is the all-zero configuration. This monoid is a group exactly when the underlying graph is a DAG (directed acyclic graph).

The recurrent elements form a submonoid which turns out to be a group. This group is called the sandpile group for $$\Gamma$$, denoted $$\mathcal{S}(\Gamma)$$. Its identity element is usually not the all-zero configuration (again, only in the case that $$\Gamma$$ is a DAG). So finding the identity element is an interesting problem.

Let $$n=|V|-1$$ and fix an ordering of the nonsink vertices. Let $$\mathcal{\tilde{L}}\subset\ZZ^n$$ denote the column-span of $$\tilde{L}^t$$, the transpose of the reduced Laplacian. It is a theorem that

$\mathcal{S}(\Gamma)\approx \ZZ^n/\mathcal{\tilde{L}}.$

Thus, the number of elements of the sandpile group is $$\det{\tilde{L}}$$, which by the matrix-tree theorem is the number of weighted trees directed into the sink.

Example. (Continued.)

sage: S.group_order()
3
sage: S.invariant_factors()
[1, 1, 3]
sage: S.reduced_laplacian().dense_matrix().smith_form()
(
[1 0 0]  [ 0  0  1]  [3 1 4]
[0 1 0]  [ 1  0  0]  [4 1 6]
[0 0 3], [ 0  1 -1], [4 1 5]
)


Adding the identity to any recurrent configuration and stabilizing yields the same recurrent configuration:

sage: S.identity()
{1: 3, 2: 1, 3: 0}
sage: i = S.identity()
sage: m = S.max_stable()
sage: i & m == m
True


Self-organized criticality¶

The sandpile model was introduced by Bak, Tang, and Wiesenfeld in the paper, Self-organized criticality: an explanation of 1/ƒ noise [BTW]. The term self-organized criticality has no precise definition, but can be loosely taken to describe a system that naturally evolves to a state that is barely stable and such that the instabilities are described by a power law. In practice, self-organized criticality is often taken to mean like the sandpile model on a grid graph. The grid graph is just a grid with an extra sink vertex. The vertices on the interior of each side have one edge to the sink, and the corner vertices have an edge of weight $$2$$. Thus, every nonsink vertex has out-degree $$4$$.

Imagine repeatedly dropping grains of sand on an empty grid graph, allowing the sandpile to stabilize in between. At first there is little activity, but as time goes on, the size and extent of the avalanche caused by a single grain of sand becomes hard to predict. Computer experiments—I do not think there is a proof, yet—indicate that the distribution of avalanche sizes obeys a power law with exponent -1. In the example below, the size of an avalanche is taken to be the sum of the number of times each vertex fires.

Example (distribution of avalanche sizes).

sage: S = sandpiles.Grid(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(10000):  # long time (15s on sage.math, 2012)
....:     m, f = m.stabilize(true)
....:     a.append(sum(f.values()))
...
sage: p = list_plot([[log(i+1),log(a.count(i))] for i in [0..max(a)] if a.count(i)])  # long time
sage: p.axes_labels(['log(N)','log(D(N))'])  # long time
sage: p  # long time
Graphics object consisting of 1 graphics primitive


Distribution of avalanche sizes

Note: In the above code, m.stabilize(true) returns a list consisting of the stabilized configuration and the firing vector. (Omitting true would give just the stabilized configuration.)

Divisors and Discrete Riemann surfaces¶

A reference for this section is Riemann-Roch and Abel-Jacobi theory on a finite graph [BN].

A divisor on $$\Gamma$$ is an element of the free abelian group on its vertices, including the sink. Suppose, as above, that the $$n+1$$ vertices of $$\Gamma$$ have been ordered, and that $$\mathcal{L}$$ is the column span of the transpose of the Laplacian. A divisor is then identified with an element $$D\in\ZZ^{n+1}$$, and two divisors are linearly equivalent if they differ by an element of $$\mathcal{L}$$. A divisor $$E$$ is effective, written $$E\geq0$$, if $$E(v)\geq0$$ for each $$v\in V$$, i.e., if $$E\in\mathbb{Z}_{\geq 0}^{n+1}$$. The degree of a divisor $$D$$ is $$\deg(D) := \sum_{v\in V} D(v)$$. The divisors of degree zero modulo linear equivalence form the Picard group, or Jacobian of the graph. For an undirected graph, the Picard group is isomorphic to the sandpile group.

The complete linear system for a divisor $$D$$, denoted $$|D|$$, is the collection of effective divisors linearly equivalent to $$D$$.

Riemann-Roch¶

To describe the Riemann-Roch theorem in this context, suppose that $$\Gamma$$ is an undirected, unweighted graph. The dimension, $$r(D)$$, of the linear system $$|D|$$ is $$-1$$ if $$|D|=\emptyset$$, and otherwise is the greatest integer $$s$$ such that $$|D-E|\neq0$$ for all effective divisors $$E$$ of degree $$s$$. Define the canonical divisor by $$K = \sum_{v\in V} (\deg(v)-2) v$$, and the genus by $$g = \#(E) - \#(V) + 1$$. The Riemann-Roch theorem says that for any divisor $$D$$,

$r(D) - r(K-D) = \deg(D) + 1 - g.$

Example.

sage: G = sandpiles.Complete(5)  # the sandpile on the complete graph with 5 vertices


A divisor on the graph:

sage: D = SandpileDivisor(G, [1,2,2,0,2])


Verify the Riemann-Roch theorem:

sage: K = G.canonical_divisor()
sage: D.rank() - (K - D).rank() == D.deg() + 1 - G.genus()
True


The effective divisors linearly equivalent to $$D$$:

sage: D.effective_div(False)
[[0, 1, 1, 4, 1], [1, 2, 2, 0, 2], [4, 0, 0, 3, 0]]


The nonspecial divisors up to linear equivalence (divisors of degree $$g-1$$ with empty linear systems):

sage: N = G.nonspecial_divisors()
sage: [E.values() for E in N[:5]]   # the first few
[[-1, 0, 1, 2, 3],
[-1, 0, 1, 3, 2],
[-1, 0, 2, 1, 3],
[-1, 0, 2, 3, 1],
[-1, 0, 3, 1, 2]]
sage: len(N)
24
sage: len(N) == G.h_vector()[-1]
True


Picturing linear systems¶

Fix a divisor $$D$$. There are at least two natural graphs associated with the linear system associated with $$D$$. First, consider the directed graph with vertex set $$|D|$$ and with an edge from vertex $$E$$ to vertex $$F$$ if $$F$$ is attained from $$E$$ by firing a single unstable vertex.

sage: S = Sandpile(graphs.CycleGraph(6),0)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: D.is_alive()
True
sage: eff = D.effective_div()
sage: firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01,iterations=500)


Complete linear system for $$(1,1,1,1,2,0)$$ on $$C_6$$: single firings

The is_alive method checks whether the divisor $$D$$ is alive, i.e., whether every divisor linearly equivalent to $$D$$ is unstable.

The second graph has the same set of vertices but with an edge from $$E$$ to $$F$$ if $$F$$ is obtained from $$E$$ by firing all unstable vertices of $$E$$.

sage: S = Sandpile(graphs.CycleGraph(6),0)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div()
sage: parallel_firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01,iterations=500)


Complete linear system for $$(1,1,1,1,2,0)$$ on $$C_6$$: parallel firings

Note that in each of the examples above, starting at any divisor in the linear system and following edges, one is eventually led into a cycle of length $$6$$ (cycling the divisor $$(1,1,1,1,2,0)$$). Thus, D.alive() returns True. In Sage, one would be able to rotate the above figures to get a better idea of the structure.

Algebraic geometry of sandpiles¶

Affine¶

Let $$n = |V| - 1$$, and fix an ordering on the nonsink vertices of $$\Gamma$$. Let $$\tilde{\mathcal{L}} \subset \ZZ^n$$ denote the column-span of $$\tilde{L}^t$$, the transpose of the reduced Laplacian. Label vertex $$i$$ with the indeterminate $$x_i$$, and let $$\CC[\Gamma_s] = \CC[x_1, \dots, x_n]$$. (Here, $$s$$ denotes the sink vertex of $$\Gamma$$.) The sandpile ideal or toppling ideal, first studied by Cori, Rossin, and Salvy [CRS] for undirected graphs, is the lattice ideal for $$\tilde{\mathcal{L}}$$:

$I = I(\Gamma_s) := \CC[\Gamma_s] \cdot \{x^u-x^v : u-v \in \tilde{\mathcal{L}}\}\subset\CC[\Gamma_s],$

where $$x^u := \prod_{i=1}^n x^{u_i}$$ for $$u \in \ZZ^n$$.

For each $$c\in\ZZ^n$$ define $$t(c) = x^{c^+} - x^{c^-}$$ where $$c^+_i = \max\{c_i,0\}$$ and $$c^-_i = \max\{-c_i,0\}$$, so that $$c = c^+ - c^-$$. Then, for each $$\sigma \in \ZZ^n$$, define $$T(\sigma) = t(\tilde{L}^t\sigma)$$. It then turns out that

$I = (T(e_1),\dots,T(e_n),x^b-1),$

where $$e_i$$ is the $$i$$-th standard basis vector and $$b$$ is any burning configuration.

The affine coordinate ring, $$\CC[\Gamma_s]/I,$$ is isomorphic to the group algebra of the sandpile group, $$\CC[\mathcal{S}(\Gamma)]$$.

The standard term-ordering on $$\CC[\Gamma_s]$$ is graded reverse lexicographical order with $$x_i > x_j$$ if vertex $$v_i$$ is further from the sink than vertex $$v_j$$. (There are choices to be made for vertices equidistant from the sink.) If $$\sigma_b$$ is the script for a burning configuration (not necessarily minimal), then

$\{T(\sigma) : \sigma \leq \sigma_b\}$

is a Gröbner basis for $$I$$.

Projective¶

Now let $$\CC[\Gamma] = \CC[x_0,x_1,\dots,x_n]$$, where $$x_0$$ corresponds to the sink vertex. The homogeneous sandpile ideal, denoted $$I^h$$, is obtaining by homogenizing $$I$$ with respect to $$x_0$$. Let $$L$$ be the (full) Laplacian, and $$\mathcal{L} \subset \ZZ^{n+1}$$ be the column span of its transpose, $$L^t.$$ Then $$I^h$$ is the lattice ideal for $$\mathcal{L}$$:

$I^h = I^h(\Gamma) := \CC[\Gamma] \cdot \{x^u-x^v: u-v \in\mathcal{L}\} \subset \CC[\Gamma].$

This ideal can be calculated by saturating the ideal

$(T(e_i): i=0, \ldots, n)$

with respect to the product of the indeterminates: $$\prod_{i=0}^n x_i$$ (extending the $$T$$ operator in the obvious way). A Gröbner basis with respect to the degree lexicographic order described above (with $$x_0$$ the smallest vertex) is obtained by homogenizing each element of the Gröbner basis for the non-homogeneous sandpile ideal with respect to $$x_0$$.

Example.

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1},
....:      3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: S = Sandpile(g, 0)
sage: S.ring()
Multivariate Polynomial Ring in x5, x4, x3, x2, x1, x0 over Rational Field


The homogeneous sandpile ideal:

sage: S.ideal()
Ideal (x2 - x0, x3^2 - x5*x0, x5*x3 - x0^2, x4^2 - x3*x1, x5^2 - x3*x0,
x1^3 - x4*x3*x0, x4*x1^2 - x5*x0^2) of Multivariate Polynomial Ring
in x5, x4, x3, x2, x1, x0 over Rational Field


The generators of the ideal:

sage: S.ideal(true)
[x2 - x0,
x3^2 - x5*x0,
x5*x3 - x0^2,
x4^2 - x3*x1,
x5^2 - x3*x0,
x1^3 - x4*x3*x0,
x4*x1^2 - x5*x0^2]


Its resolution:

sage: S.resolution() # long time
'R^1 <-- R^7 <-- R^19 <-- R^25 <-- R^16 <-- R^4'


and Betti table:

sage: S.betti() # long time
0     1     2     3     4     5
------------------------------------------
0:     1     1     -     -     -     -
1:     -     4     6     2     -     -
2:     -     2     7     7     2     -
3:     -     -     6    16    14     4
------------------------------------------
total:     1     7    19    25    16     4


The Hilbert function:

sage: S.hilbert_function()
[1, 5, 11, 15]


and its first differences (which count the number of superstable configurations in each degree):

sage: S.h_vector()
[1, 4, 6, 4]
sage: x = [i.deg() for i in S.superstables()]
sage: sorted(x)
[0, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3]


The degree in which the Hilbert function starts equalling the Hilbert polynomial, the latter always being a constant in the case of a sandpile ideal:

sage: S.postulation()
3


Zeros¶

The zero set for the sandpile ideal $$I$$ is

$Z(I) = \{ p\in\CC^n : f(p)=0\mbox{ for all f\in I} \},$

the set of simultaneous zeros of the polynomials in $$I$$. Letting $$S^1$$ denote the unit circle in the complex plane, $$Z(I)$$ is a finite subgroup of $$S^1 \times \cdots \times S^1 \subset \CC^n$$, isomorphic to the sandpile group. The zero set is actually linearly isomorphic to a faithful representation of the sandpile group on $$\CC^n$$.

Todo

The above is not quite true. $$Z(I)$$ is neither finite nor a subgroup of $$S^1 \times \cdots \times S^1 \subset \CC^n$$. What is probably meant is that the subset of $$Z(I)$$ in which the coordinate of $$p$$ corresponding to the sink is set to $$1$$ is a finite subgroup of $$S^1 \times \cdots \times S^1 \subset \CC^{n-1}$$.

Example. (Continued.)

sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0)
sage: S.ideal().gens()
[x1^2 - x2^2, x1*x2^3 - x0^4, x2^5 - x1*x0^4]


Approximation to the zero set (setting x_0 = 1):

sage: S.solve()
[[-0.707107 + 0.707107*I, 0.707107 - 0.707107*I],
[-0.707107 - 0.707107*I, 0.707107 + 0.707107*I],
[-I, -I],
[I, I],
[0.707107 + 0.707107*I, -0.707107 - 0.707107*I],
[0.707107 - 0.707107*I, -0.707107 + 0.707107*I],
[1, 1],
[-1, -1]]
sage: len(_) == S.group_order()
True


The zeros are generated as a group by a single vector:

sage: S.points()
[[(1/2*I + 1/2)*sqrt(2), -(1/2*I + 1/2)*sqrt(2)]]


Resolutions¶

The homogeneous sandpile ideal, $$I^h$$, has a free resolution graded by the divisors on $$\Gamma$$ modulo linear equivalence. (See the section on Discrete Riemann Surfaces for the language of divisors and linear equivalence.) Let $$S = \CC[\Gamma] = \CC[x_0,\dots,x_n]$$, as above, and let $$\mathfrak{S}$$ denote the group of divisors modulo rational equivalence. Then $$S$$ is graded by $$\mathfrak{S}$$ by letting $$\deg(x^c) = c \in \mathfrak{S}$$ for each monomial $$x^c$$. The minimal free resolution of $$I^h$$ has the form

$0 \leftarrow I^h \leftarrow \bigoplus_{D\in\mathfrak{S}} S(-D)^{\beta_{0,D}} \leftarrow \bigoplus_{D\in\mathfrak{S}} S(-D)^{\beta_{1,D}} \leftarrow \cdots \leftarrow \bigoplus_{D\in\mathfrak{S}} S(-D)^{\beta_{r,D}} \leftarrow 0,$

where the $$\beta_{i,D}$$ are the Betti numbers for $$I^h$$.

For each divisor class $$D\in\mathfrak{S}$$, define a simplicial complex

$\Delta_D := \{I\subseteq\{0,\dots,n\}: I\subseteq\mbox{supp}(E) \mbox{ for some } E \in |D|\}.$

The Betti number $$\beta_{i,D}$$ equals the dimension over $$\CC$$ of the $$i$$-th reduced homology group of $$\Delta_D$$:

$\beta_{i,D} = \dim_{\CC}\tilde{H}_i(\Delta_D;\CC).$
sage: S = Sandpile({0:{},1:{0: 1, 2: 1, 3: 4},2:{3: 5},3:{1: 1, 2: 1}},0)


Representatives of all divisor classes with nontrivial homology:

sage: p = S.betti_complexes()
sage: p[0]
[{0: -8, 1: 5, 2: 4, 3: 1},
Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}]


The homology associated with the first divisor in the list:

sage: D = p[0][0]
sage: D.effective_div()
[{0: 0, 1: 0, 2: 0, 3: 2}, {0: 0, 1: 1, 2: 1, 3: 0}]
sage: [E.support() for E in D.effective_div()]
[[3], [1, 2]]
sage: D.Dcomplex()
Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}
sage: D.Dcomplex().homology()
{0: Z, 1: 0}


The minimal free resolution:

sage: S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'
sage: S.betti()
0     1     2     3
------------------------------
0:     1     -     -     -
1:     -     5     5     -
2:     -     -     -     1
------------------------------
total:     1     5     5     1
sage: len(p)
11


The degrees and ranks of the homology groups for each element of the list p (compare with the Betti table, above):

sage: [[sum(d[0].values()),d[1].betti()] for d in p]
[[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1}],
[2, {0: 2, 1: 0}],
[3, {0: 1, 1: 1, 2: 0}],
[5, {0: 1, 1: 0, 2: 1}]]


Complete Intersections and Arithmetically Gorenstein toppling ideals¶

NOTE: in the previous section note that the resolution always has length $$n$$ since the ideal is Cohen-Macaulay.

To do.

To do.

Usage¶

Initialization¶

There are three main classes for sandpile structures in Sage: Sandpile, SandpileConfig, and SandpileDivisor. Initialization for Sandpile has the form

.. skip

sage: S = Sandpile(graph, sink)

where graph represents a graph and sink is the key for the sink vertex. There are four possible forms for graph:

1. a Python dictionary of dictionaries:

sage: g = {0: {}, 1: {0: 1, 3: 1, 4: 1}, 2: {0: 1, 3: 1, 5: 1},
....:      3: {2: 1, 5: 1}, 4: {1: 1, 3: 1}, 5: {2: 1, 3: 1}}


Graph from dictionary of dictionaries.

Each key is the name of a vertex. Next to each vertex name $$v$$ is a dictionary consisting of pairs: vertex: weight. Each pair represents a directed edge emanating from $$v$$ and ending at vertex having (non-negative integer) weight equal to weight. Loops are allowed. In the example above, all of the weights are 1.

2. a Python dictionary of lists:

sage: g = {0: [], 1: [0, 3, 4], 2: [0, 3, 5],
....:      3: [2, 5], 4: [1, 3], 5: [2, 3]}


This is a short-hand when all of the edge-weights are equal to 1. The above example is for the same displayed graph.

3. a Sage graph (of type sage.graphs.graph.Graph):

sage: g = graphs.CycleGraph(5)
sage: S = Sandpile(g, 0)
sage: type(g)
<class 'sage.graphs.graph.Graph'>


To see the types of built-in graphs, type graphs., including the period, and hit TAB.

4. a Sage digraph:

sage: S = Sandpile(digraphs.RandomDirectedGNC(6), 0)
sage: S.show()


A random graph.

See sage.graphs.graph_generators for more information on the Sage graph library and graph constructors.

Each of these four formats is preprocessed by the Sandpile class so that, internally, the graph is represented by the dictionary of dictionaries format first presented. This internal format is returned by dict():

sage: S = Sandpile({0:[], 1:[0, 3, 4], 2:[0, 3, 5], 3: [2, 5], 4: [1, 3], 5: [2, 3]},0)
sage: S.dict()
{0: {},
1: {0: 1, 3: 1, 4: 1},
2: {0: 1, 3: 1, 5: 1},
3: {2: 1, 5: 1},
4: {1: 1, 3: 1},
5: {2: 1, 3: 1}}


Note

The user is responsible for assuring that each vertex has a directed path into the designated sink. If the sink has out-edges, these will be ignored for the purposes of sandpile calculations (but not calculations on divisors).

Code for checking whether a given vertex is a sink:

sage: S = Sandpile({0:[], 1:[0, 3, 4], 2:[0, 3, 5], 3: [2, 5], 4: [1, 3], 5: [2, 3]},0)
sage: [S.distance(v,0) for v in S.vertices()] # 0 is a sink
[0, 1, 1, 2, 2, 2]
sage: [S.distance(v,1) for v in S.vertices()] # 1 is not a sink
[+Infinity, 0, +Infinity, +Infinity, 1, +Infinity]


Methods¶

Here are summaries of Sandpile, SandpileConfig, and SandpileDivisor methods (functions). Each summary is followed by a list of complete descriptions of the methods. There are many more methods available for a Sandpile, e.g., those inherited from the class DiGraph. To see them all, enter dir(Sandpile) or type Sandpile., including the period, and hit TAB.

Sandpile¶

Summary of methods.

• all_k_config — The constant configuration with all values set to k.
• all_k_div — The divisor with all values set to k.
• avalanche_polynomial — The avalanche polynomial.
• betti — The Betti table for the homogeneous toppling ideal.
• betti_complexes — The support-complexes with non-trivial homology.
• burning_config — The minimal burning configuration.
• burning_script — A script for the minimal burning configuration.
• canonical_divisor — The canonical divisor.
• dict — A dictionary of dictionaries representing a directed graph.
• genus — The genus: (# non-loop edges) - (# vertices) + 1.
• groebner — A Groebner basis for the homogeneous toppling ideal.
• group_gens — A minimal list of generators for the sandpile group.
• group_order — The size of the sandpile group.
• h_vector — The number of superstable configurations in each degree.
• help — List of Sandpile-specific methods (not inherited from Graph).
• hilbert_function — The Hilbert function of the homogeneous toppling ideal.
• ideal — The saturated homogeneous toppling ideal.
• identity — The identity configuration.
• in_degree — The in-degree of a vertex or a list of all in-degrees.
• invariant_factors — The invariant factors of the sandpile group.
• is_undirected — Is the underlying graph undirected?
• jacobian_representatives — Representatives for the elements of the Jacobian group.
• laplacian — The Laplacian matrix of the graph.
• markov_chain — The sandpile Markov chain for configurations or divisors.
• max_stable — The maximal stable configuration.
• max_stable_div — The maximal stable divisor.
• max_superstables — The maximal superstable configurations.
• min_recurrents — The minimal recurrent elements.
• nonsink_vertices — The nonsink vertices.
• nonspecial_divisors — The nonspecial divisors.
• out_degree — The out-degree of a vertex or a list of all out-degrees.
• picard_representatives — Representatives of the divisor classes of degree d in the Picard group.
• points — Generators for the multiplicative group of zeros of the sandpile ideal.
• postulation — The postulation number of the toppling ideal.
• recurrents — The recurrent configurations.
• reduced_laplacian — The reduced Laplacian matrix of the graph.
• reorder_vertices — A copy of the sandpile with vertex names permuted.
• resolution — A minimal free resolution of the homogeneous toppling ideal.
• ring — The ring containing the homogeneous toppling ideal.
• show — Draw the underlying graph.
• show3d — Draw the underlying graph.
• sink — The sink vertex.
• smith_form — The Smith normal form for the Laplacian.
• solve — Approximations of the complex affine zeros of the sandpile ideal.
• stable_configs — Generator for all stable configurations.
• stationary_density — The stationary density of the sandpile.
• superstables — The superstable configurations.
• symmetric_recurrents — The symmetric recurrent configurations.
• tutte_polynomial — The Tutte polynomial.
• unsaturated_ideal — The unsaturated, homogeneous toppling ideal.
• version — The version number of Sage Sandpiles.
• zero_config — The all-zero configuration.
• zero_div — The all-zero divisor.

Complete descriptions of Sandpile methods.

all_k_config(k)

The constant configuration with all values set to $$k$$.

INPUT:

k – integer

OUTPUT:

SandpileConfig

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.all_k_config(7)
{1: 7, 2: 7, 3: 7}


all_k_div(k)

The divisor with all values set to $$k$$.

INPUT:

k – integer

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.House()
sage: S.all_k_div(7)
{0: 7, 1: 7, 2: 7, 3: 7, 4: 7}


avalanche_polynomial(multivariable=True)

The avalanche polynomial. See NOTE for details.

INPUT:

multivariable – (default: True) boolean

OUTPUT:

polynomial

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: s.avalanche_polynomial()
9*x0*x1*x2 + 2*x0*x1 + 2*x0*x2 + 2*x1*x2 + 3*x0 + 3*x1 + 3*x2 + 24
sage: s.avalanche_polynomial(False)
9*x0^3 + 6*x0^2 + 9*x0 + 24


Note

For each nonsink vertex $$v$$, let $$x_v$$ be an indeterminate. If $$(r,v)$$ is a pair consisting of a recurrent $$r$$ and nonsink vertex $$v$$, then for each nonsink vertex $$w$$, let $$n_w$$ be the number of times vertex $$w$$ fires in the stabilization of $$r + v$$. Let $$M(r,v)$$ be the monomial $$\prod_w x_w^{n_w}$$, i.e., the exponent records the vector of $$n_w$$ as $$w$$ ranges over the nonsink vertices. The avalanche polynomial is then the sum of $$M(r,v)$$ as $$r$$ ranges over the recurrents and $$v$$ ranges over the nonsink vertices. If multivariable is False, then set all the indeterminates equal to each other (and, thus, only count the number of vertex firings in the stabilizations, forgetting which particular vertices fired).

betti(verbose=True)

The Betti table for the homogeneous toppling ideal. If verbose is True, it prints the standard Betti table, otherwise, it returns a less formatted table.

INPUT:

verbose – (default: True) boolean

OUTPUT:

Betti numbers for the sandpile

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.betti()
0     1     2     3
------------------------------
0:     1     -     -     -
1:     -     2     -     -
2:     -     4     9     4
------------------------------
total:     1     6     9     4
sage: S.betti(False)
[1, 6, 9, 4]


betti_complexes()

The support-complexes with non-trivial homology. (See NOTE.)

OUTPUT:

list (of pairs [divisors, corresponding simplicial complex])

EXAMPLES:

sage: S = Sandpile({0:{},1:{0: 1, 2: 1, 3: 4},2:{3: 5},3:{1: 1, 2: 1}},0)
sage: p = S.betti_complexes()
sage: p[0]
[{0: -8, 1: 5, 2: 4, 3: 1}, Simplicial complex with vertex set (1, 2, 3) and facets {(3,), (1, 2)}]
sage: S.resolution()
'R^1 <-- R^5 <-- R^5 <-- R^1'
sage: S.betti()
0     1     2     3
------------------------------
0:     1     -     -     -
1:     -     5     5     -
2:     -     -     -     1
------------------------------
total:     1     5     5     1
sage: len(p)
11
sage: p[0][1].homology()
{0: Z, 1: 0}
sage: p[-1][1].homology()
{0: 0, 1: 0, 2: Z}


Note

A support-complex is the simplicial complex formed from the supports of the divisors in a linear system.

burning_config()

The minimal burning configuration.

OUTPUT:

dict (configuration)

EXAMPLES:

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1}, \
3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: S = Sandpile(g,0)
sage: S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
sage: S.burning_config().values()
[2, 0, 1, 1, 0]
sage: S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
sage: script = S.burning_script().values()
sage: script
[1, 3, 5, 1, 4]
sage: matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]


Note

The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.

A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if $$b$$ is the burning configuration, $$\sigma$$ is its script, and $$\tilde{L}$$ is the reduced Laplacian, then $$\sigma\cdot \tilde{L} = b$$. The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).

The following are equivalent for a configuration $$c$$ with burning configuration $$b$$ having script $$\sigma$$:

• $$c$$ is recurrent;
• $$c+b$$ stabilizes to $$c$$;
• the firing vector for the stabilization of $$c+b$$ is $$\sigma$$.

burning_script()

A script for the minimal burning configuration.

OUTPUT:

dict

EXAMPLES:

sage: g = {0:{},1:{0:1,3:1,4:1},2:{0:1,3:1,5:1},\
3:{2:1,5:1},4:{1:1,3:1},5:{2:1,3:1}}
sage: S = Sandpile(g,0)
sage: S.burning_config()
{1: 2, 2: 0, 3: 1, 4: 1, 5: 0}
sage: S.burning_config().values()
[2, 0, 1, 1, 0]
sage: S.burning_script()
{1: 1, 2: 3, 3: 5, 4: 1, 5: 4}
sage: script = S.burning_script().values()
sage: script
[1, 3, 5, 1, 4]
sage: matrix(script)*S.reduced_laplacian()
[2 0 1 1 0]


Note

The burning configuration and script are computed using a modified version of Speer’s script algorithm. This is a generalization to directed multigraphs of Dhar’s burning algorithm.

A burning configuration is a nonnegative integer-linear combination of the rows of the reduced Laplacian matrix having nonnegative entries and such that every vertex has a path from some vertex in its support. The corresponding burning script gives the integer-linear combination needed to obtain the burning configuration. So if $$b$$ is the burning configuration, $$s$$ is its script, and $$L_{\mathrm{red}}$$ is the reduced Laplacian, then $$s\cdot L_{\mathrm{red}}= b$$. The minimal burning configuration is the one with the minimal script (its components are no larger than the components of any other script for a burning configuration).

The following are equivalent for a configuration $$c$$ with burning configuration $$b$$ having script $$s$$:

• $$c$$ is recurrent;
• $$c+b$$ stabilizes to $$c$$;
• the firing vector for the stabilization of $$c+b$$ is $$s$$.

canonical_divisor()

The canonical divisor. This is the divisor with $$\deg(v)-2$$ grains of sand on each vertex (not counting loops). Only for undirected graphs.

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.Complete(4)
sage: S.canonical_divisor()
{0: 1, 1: 1, 2: 1, 3: 1}
sage: s = Sandpile({0:[1,1],1:[0,0,1,1,1]},0)
sage: s.canonical_divisor()  # loops are disregarded
{0: 0, 1: 0}


Warning

The underlying graph must be undirected.

dict()

A dictionary of dictionaries representing a directed graph.

OUTPUT:

dict

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.dict()
{0: {1: 1, 2: 1},
1: {0: 1, 2: 1, 3: 1},
2: {0: 1, 1: 1, 3: 1},
3: {1: 1, 2: 1}}
sage: S.sink()
0


genus()

The genus: (# non-loop edges) - (# vertices) + 1. Only defined for undirected graphs.

OUTPUT:

integer

EXAMPLES:

sage: sandpiles.Complete(4).genus()
3
sage: sandpiles.Cycle(5).genus()
1


groebner()

A Groebner basis for the homogeneous toppling ideal. It is computed with respect to the standard sandpile ordering (see ring).

OUTPUT:

Groebner basis

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.groebner()
[x3*x2^2 - x1^2*x0, x2^3 - x3*x1*x0, x3*x1^2 - x2^2*x0, x1^3 - x3*x2*x0, x3^2 - x0^2, x2*x1 - x0^2]


group_gens(verbose=True)

A minimal list of generators for the sandpile group. If verbose is False then the generators are represented as lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileConfig (or of lists of integers if verbose is False)

EXAMPLES:

sage: s = sandpiles.Cycle(5)
sage: s.group_gens()
[{1: 1, 2: 1, 3: 1, 4: 0}]
sage: s.group_gens()[0].order()
5
sage: s = sandpiles.Complete(5)
sage: s.group_gens(False)
[[2, 2, 3, 2], [2, 3, 2, 2], [3, 2, 2, 2]]
sage: [i.order() for i in s.group_gens()]
[5, 5, 5]
sage: s.invariant_factors()
[1, 5, 5, 5]


group_order()

The size of the sandpile group.

OUTPUT:

integer

EXAMPLES:

sage: S = sandpiles.House()
sage: S.group_order()
11


h_vector()

The number of superstable configurations in each degree. Equivalently, this is the list of first differences of the Hilbert function of the (homogeneous) toppling ideal.

OUTPUT:

list of nonnegative integers

EXAMPLES:

sage: s = sandpiles.Grid(2,2)
sage: s.hilbert_function()
[1, 5, 15, 35, 66, 106, 146, 178, 192]
sage: s.h_vector()
[1, 4, 10, 20, 31, 40, 40, 32, 14]


help(verbose=True)

List of Sandpile-specific methods (not inherited from Graph). If verbose, include short descriptions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

printed string

EXAMPLES:

sage: Sandpile.help()
For detailed help with any method FOO listed below,
enter "Sandpile.FOO?" or enter "S.FOO?" for any Sandpile S.

all_k_config             -- The constant configuration with all values set to k.
all_k_div                -- The divisor with all values set to k.
avalanche_polynomial     -- The avalanche polynomial.
betti                    -- The Betti table for the homogeneous toppling ideal.
betti_complexes          -- The support-complexes with non-trivial homology.
burning_config           -- The minimal burning configuration.
burning_script           -- A script for the minimal burning configuration.
canonical_divisor        -- The canonical divisor.
dict                     -- A dictionary of dictionaries representing a directed graph.
genus                    -- The genus: (# non-loop edges) - (# vertices) + 1.
groebner                 -- A Groebner basis for the homogeneous toppling ideal.
group_gens               -- A minimal list of generators for the sandpile group.
group_order              -- The size of the sandpile group.
h_vector                 -- The number of superstable configurations in each degree.
help                     -- List of Sandpile-specific methods (not inherited from "Graph").
hilbert_function         -- The Hilbert function of the homogeneous toppling ideal.
ideal                    -- The saturated homogeneous toppling ideal.
identity                 -- The identity configuration.
in_degree                -- The in-degree of a vertex or a list of all in-degrees.
invariant_factors        -- The invariant factors of the sandpile group.
is_undirected            -- Is the underlying graph undirected?
jacobian_representatives -- Representatives for the elements of the Jacobian group.
laplacian                -- The Laplacian matrix of the graph.
markov_chain             -- The sandpile Markov chain for configurations or divisors.
max_stable               -- The maximal stable configuration.
max_stable_div           -- The maximal stable divisor.
max_superstables         -- The maximal superstable configurations.
min_recurrents           -- The minimal recurrent elements.
nonsink_vertices         -- The nonsink vertices.
nonspecial_divisors      -- The nonspecial divisors.
out_degree               -- The out-degree of a vertex or a list of all out-degrees.
picard_representatives   -- Representatives of the divisor classes of degree d in the Picard group.
points                   -- Generators for the multiplicative group of zeros of the sandpile ideal.
postulation              -- The postulation number of the toppling ideal.
recurrents               -- The recurrent configurations.
reduced_laplacian        -- The reduced Laplacian matrix of the graph.
reorder_vertices         -- A copy of the sandpile with vertex names permuted.
resolution               -- A minimal free resolution of the homogeneous toppling ideal.
ring                     -- The ring containing the homogeneous toppling ideal.
show                     -- Draw the underlying graph.
show3d                   -- Draw the underlying graph.
sink                     -- The sink vertex.
smith_form               -- The Smith normal form for the Laplacian.
solve                    -- Approximations of the complex affine zeros of the sandpile ideal.
stable_configs           -- Generator for all stable configurations.
stationary_density       -- The stationary density of the sandpile.
superstables             -- The superstable configurations.
symmetric_recurrents     -- The symmetric recurrent configurations.
tutte_polynomial         -- The Tutte polynomial of the underlying graph.
unsaturated_ideal        -- The unsaturated, homogeneous toppling ideal.
version                  -- The version number of Sage Sandpiles.
zero_config              -- The all-zero configuration.
zero_div                 -- The all-zero divisor.


hilbert_function()

The Hilbert function of the homogeneous toppling ideal.

OUTPUT:

list of nonnegative integers

EXAMPLES:

sage: s = sandpiles.Wheel(5)
sage: s.hilbert_function()
[1, 5, 15, 31, 45]
sage: s.h_vector()
[1, 4, 10, 16, 14]


ideal(gens=False)

The saturated homogeneous toppling ideal. If gens is True, the generators for the ideal are returned instead.

INPUT:

gens – (default: False) boolean

OUTPUT:

ideal or, optionally, the generators of an ideal

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.ideal()
Ideal (x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0) of Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
sage: S.ideal(True)
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]
sage: S.ideal().gens()  # another way to get the generators
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]


identity(verbose=True)

The identity configuration. If verbose is False, the configuration are converted to a list of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

SandpileConfig or a list of integers If verbose is False, the configuration are converted to a list of integers.

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.identity()
{1: 2, 2: 2, 3: 0}
sage: s.identity(False)
[2, 2, 0]
sage: s.identity() & s.max_stable() == s.max_stable()
True


in_degree(v=None)

The in-degree of a vertex or a list of all in-degrees.

INPUT:

v – (optional) vertex name

OUTPUT:

integer or dict

EXAMPLES:

sage: s = sandpiles.House()
sage: s.in_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
sage: s.in_degree(2)
3


invariant_factors()

The invariant factors of the sandpile group.

OUTPUT:

list of integers

EXAMPLES:

sage: s = sandpiles.Grid(2,2)
sage: s.invariant_factors()
[1, 1, 8, 24]


is_undirected()

Is the underlying graph undirected? True if $$(u,v)$$ is and edge if and only if $$(v,u)$$ is an edge, each edge with the same weight.

OUTPUT:

boolean

EXAMPLES:

sage: sandpiles.Complete(4).is_undirected()
True
sage: s = Sandpile({0:[1,2], 1:[0,2], 2:[0]}, 0)
sage: s.is_undirected()
False


jacobian_representatives(verbose=True)

Representatives for the elements of the Jacobian group. If verbose is False, then lists representing the divisors are returned.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileDivisor (or of lists representing divisors)

EXAMPLES:

For an undirected graph, divisors of the form s - deg(s)*sink as s varies over the superstables forms a distinct set of representatives for the Jacobian group.:

sage: s = sandpiles.Complete(3)
sage: s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
sage: s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]


If the graph is directed, the representatives described above may by equivalent modulo the rowspan of the Laplacian matrix:

sage: s = Sandpile({0: {1: 1, 2: 2}, 1: {0: 2, 2: 4}, 2: {0: 4, 1: 2}},0)
sage: s.group_order()
28
sage: s.jacobian_representatives()
[{0: -5, 1: 3, 2: 2}, {0: -4, 1: 3, 2: 1}]


Let $$\tau$$ be the nonnegative generator of the kernel of the transpose of the Laplacian, and let $$tau_s$$ be it sink component, then the sandpile group is isomorphic to the direct sum of the cyclic group of order $$\tau_s$$ and the Jacobian group. In the example above, we have:

sage: s.laplacian().left_kernel()
Free module of degree 3 and rank 1 over Integer Ring
Echelon basis matrix:
[14  5  8]


Note

The Jacobian group is the set of all divisors of degree zero modulo the integer rowspan of the Laplacian matrix.

laplacian()

The Laplacian matrix of the graph. Its rows encode the vertex firing rules.

OUTPUT:

matrix

EXAMPLES:

sage: G = sandpiles.Diamond()
sage: G.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]


Warning

The function laplacian_matrix should be avoided. It returns the indegree version of the Laplacian.

markov_chain(state, distrib=None)

The sandpile Markov chain for configurations or divisors. The chain starts at state. See NOTE for details.

INPUT:

• state – SandpileConfig, SandpileDivisor, or list representing one of these
• distrib – (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)

OUTPUT:

generator for Markov chain (see NOTE)

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: m = s.markov_chain([0,0,0])
sage: next(m)          # random
{1: 0, 2: 0, 3: 0}
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 0, 0]
sage: next(m).values() # random
[0, 1, 0]
sage: next(m).values() # random
[0, 2, 0]
sage: next(m).values() # random
[0, 2, 1]
sage: next(m).values() # random
[1, 2, 1]
sage: next(m).values() # random
[2, 2, 1]
sage: m = s.markov_chain(s.zero_div(), [0.1,0.1,0.1,0.7])
sage: next(m).values() # random
[0, 0, 0, 1]
sage: next(m).values() # random
[0, 0, 1, 1]
sage: next(m).values() # random
[0, 0, 1, 2]
sage: next(m).values() # random
[1, 1, 2, 0]
sage: next(m).values() # random
[1, 1, 2, 1]
sage: next(m).values() # random
[1, 1, 2, 2]
sage: next(m).values() # random
[1, 1, 2, 3]
sage: next(m).values() # random
[1, 1, 2, 4]
sage: next(m).values() # random
[1, 1, 3, 4]


Note

The closed sandpile Markov chain has state space consisting of the configurations on a sandpile. It transitions from a state by choosing a vertex at random (according to the probability distribution distrib), dropping a grain of sand at that vertex, and stabilizing. If the chosen vertex is the sink, the chain stays at the current state.

The open sandpile Markov chain has state space consisting of the recurrent elements, i.e., the state space is the sandpile group. It transitions from the configuration $$c$$ by choosing a vertex $$v$$ at random according to distrib. The next state is the stabilization of $$c+v$$. If $$v$$ is the sink vertex, then the stabilization of $$c+v$$ is defined to be $$c$$.

Note that in either case, if distrib is specified, its length is equal to the total number of vertices (including the sink).

REFERENCES:

 [Levine2014] (1, 2) Lionel Levine. Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle, Communications in Mathematical Physics. Arxiv 1402.3283

max_stable()

The maximal stable configuration.

OUTPUT:

SandpileConfig (the maximal stable configuration)

EXAMPLES:

sage: S = sandpiles.House()
sage: S.max_stable()
{1: 1, 2: 2, 3: 2, 4: 1}


max_stable_div()

The maximal stable divisor.

OUTPUT:

SandpileDivisor (the maximal stable divisor)

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.max_stable_div()
{0: 1, 1: 2, 2: 2, 3: 1}
sage: s.out_degree()
{0: 2, 1: 3, 2: 3, 3: 2}


max_superstables(verbose=True)

The maximal superstable configurations. If the underlying graph is undirected, these are the superstables of highest degree. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

tuple of SandpileConfig

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.superstables(False)
[[0, 0, 0],
[0, 0, 1],
[1, 0, 1],
[0, 2, 0],
[2, 0, 0],
[0, 1, 1],
[1, 0, 0],
[0, 1, 0]]
sage: s.max_superstables(False)
[[1, 0, 1], [0, 2, 0], [2, 0, 0], [0, 1, 1]]
sage: s.h_vector()
[1, 3, 4]


min_recurrents(verbose=True)

The minimal recurrent elements. If the underlying graph is undirected, these are the recurrent elements of least degree. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileConfig

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.recurrents(False)
[[2, 2, 1],
[2, 2, 0],
[1, 2, 0],
[2, 0, 1],
[0, 2, 1],
[2, 1, 0],
[1, 2, 1],
[2, 1, 1]]
sage: s.min_recurrents(False)
[[1, 2, 0], [2, 0, 1], [0, 2, 1], [2, 1, 0]]
sage: [i.deg() for i in s.recurrents()]
[5, 4, 3, 3, 3, 3, 4, 4]


nonsink_vertices()

The nonsink vertices.

OUTPUT:

list of vertices

EXAMPLES:

sage: s = sandpiles.Grid(2,3)
sage: s.nonsink_vertices()
[(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)]


nonspecial_divisors(verbose=True)

The nonspecial divisors. Only for undirected graphs. (See NOTE.)

INPUT:

verbose – (default: True) boolean

OUTPUT:

list (of divisors)

EXAMPLES:

sage: S = sandpiles.Complete(4)
sage: ns = S.nonspecial_divisors()
sage: D = ns[0]
sage: D.values()
[-1, 0, 1, 2]
sage: D.deg()
2
sage: [i.effective_div() for i in ns]
[[], [], [], [], [], []]


Note

The “nonspecial divisors” are those divisors of degree $$g-1$$ with empty linear system. The term is only defined for undirected graphs. Here, $$g = |E| - |V| + 1$$ is the genus of the graph (not counting loops as part of $$|E|$$). If verbose is False, the divisors are converted to lists of integers.

Warning

The underlying graph must be undirected.

out_degree(v=None)

The out-degree of a vertex or a list of all out-degrees.

INPUT:

v - (optional) vertex name

OUTPUT:

integer or dict

EXAMPLES:

sage: s = sandpiles.House()
sage: s.out_degree()
{0: 2, 1: 2, 2: 3, 3: 3, 4: 2}
sage: s.out_degree(2)
3


picard_representatives(d, verbose=True)

Representatives of the divisor classes of degree $$d$$ in the Picard group. (Also see the documentation for jacobian_representatives.)

INPUT:

• d – integer
• verbose – (default: True) boolean

OUTPUT:

list of SandpileDivisors (or lists representing divisors)

EXAMPLES:

sage: s = sandpiles.Complete(3)
sage: s.superstables(False)
[[0, 0], [0, 1], [1, 0]]
sage: s.jacobian_representatives(False)
[[0, 0, 0], [-1, 0, 1], [-1, 1, 0]]
sage: s.picard_representatives(3,False)
[[3, 0, 0], [2, 0, 1], [2, 1, 0]]


points()

Generators for the multiplicative group of zeros of the sandpile ideal.

OUTPUT:

list of complex numbers

EXAMPLES:

The sandpile group in this example is cyclic, and hence there is a single generator for the group of solutions.

sage: S = sandpiles.Complete(4)
sage: S.points()
[[1, I, -I], [I, 1, -I]]


postulation()

The postulation number of the toppling ideal. This is the largest weight of a superstable configuration of the graph.

OUTPUT:

nonnegative integer

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: s.postulation()
3


recurrents(verbose=True)

The recurrent configurations. If verbose is False, the configurations are converted to lists of integers.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of recurrent configurations

EXAMPLES:

sage: r = Sandpile(graphs.HouseXGraph(),0).recurrents()
sage: r[:3]
[{1: 2, 2: 3, 3: 3, 4: 1}, {1: 1, 2: 3, 3: 3, 4: 0}, {1: 1, 2: 3, 3: 3, 4: 1}]
sage: sandpiles.Complete(4).recurrents(False)
[[2, 2, 2],
[2, 2, 1],
[2, 1, 2],
[1, 2, 2],
[2, 2, 0],
[2, 0, 2],
[0, 2, 2],
[2, 1, 1],
[1, 2, 1],
[1, 1, 2],
[2, 1, 0],
[2, 0, 1],
[1, 2, 0],
[1, 0, 2],
[0, 2, 1],
[0, 1, 2]]
sage: sandpiles.Cycle(4).recurrents(False)
[[1, 1, 1], [0, 1, 1], [1, 0, 1], [1, 1, 0]]


reduced_laplacian()

The reduced Laplacian matrix of the graph.

OUTPUT:

matrix

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.laplacian()
[ 2 -1 -1  0]
[-1  3 -1 -1]
[-1 -1  3 -1]
[ 0 -1 -1  2]
sage: S.reduced_laplacian()
[ 3 -1 -1]
[-1  3 -1]
[-1 -1  2]


Note

This is the Laplacian matrix with the row and column indexed by the sink vertex removed.

reorder_vertices()

A copy of the sandpile with vertex names permuted. After reordering, vertex $$u$$ comes before vertex $$v$$ in the list of vertices if $$u$$ is closer to the sink.

OUTPUT:

Sandpile

EXAMPLES:

sage: S = Sandpile({0:[1], 2:[0,1], 1:[2]})
sage: S.dict()
{0: {1: 1}, 1: {2: 1}, 2: {0: 1, 1: 1}}
sage: T = S.reorder_vertices()


The vertices 1 and 2 have been swapped:

sage: T.dict()
{0: {1: 1}, 1: {0: 1, 2: 1}, 2: {0: 1}}


resolution(verbose=False)

A minimal free resolution of the homogeneous toppling ideal. If verbose is True, then all of the mappings are returned. Otherwise, the resolution is summarized.

INPUT:

verbose – (default: False) boolean

OUTPUT:

free resolution of the toppling ideal

EXAMPLES:

sage: S = Sandpile({0: {}, 1: {0: 1, 2: 1, 3: 4}, 2: {3: 5}, 3: {1: 1, 2: 1}},0)
sage: S.resolution()  # a Gorenstein sandpile graph
'R^1 <-- R^5 <-- R^5 <-- R^1'
sage: S.resolution(True)
[
[ x1^2 - x3*x0 x3*x1 - x2*x0  x3^2 - x2*x1  x2*x3 - x0^2  x2^2 - x1*x0],

[ x3  x2   0  x0   0]  [ x2^2 - x1*x0]
[-x1 -x3  x2   0 -x0]  [-x2*x3 + x0^2]
[ x0  x1   0  x2   0]  [-x3^2 + x2*x1]
[  0   0 -x1 -x3  x2]  [x3*x1 - x2*x0]
[  0   0  x0  x1 -x3], [ x1^2 - x3*x0]
]
sage: r = S.resolution(True)
sage: r[0]*r[1]
[0 0 0 0 0]
sage: r[1]*r[2]
[0]
[0]
[0]
[0]
[0]


ring()

The ring containing the homogeneous toppling ideal.

OUTPUT:

ring

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.ring()
Multivariate Polynomial Ring in x3, x2, x1, x0 over Rational Field
sage: S.ring().gens()
(x3, x2, x1, x0)


Note

The indeterminate xi corresponds to the $$i$$-th vertex as listed my the method vertices. The term-ordering is degrevlex with indeterminates ordered according to their distance from the sink (larger indeterminates are further from the sink).

show(**kwds)

Draw the underlying graph.

INPUT:

kwds – (optional) arguments passed to the show method for Graph or DiGraph

EXAMPLES:

sage: S = Sandpile({0:[], 1:[0,3,4], 2:[0,3,5], 3:[2,5], 4:[1,1], 5:[2,4]})
sage: S.show()
sage: S.show(graph_border=True, edge_labels=True)


show3d(**kwds)

Draw the underlying graph.

INPUT:

kwds – (optional) arguments passed to the show method for Graph or DiGraph

EXAMPLES:

sage: S = sandpiles.House()
sage: S.show3d()


sink()

The sink vertex.

OUTPUT:

sink vertex

EXAMPLES:

sage: G = sandpiles.House()
sage: G.sink()
0
sage: H = sandpiles.Grid(2,2)
sage: H.sink()
(0, 0)
sage: type(H.sink())
<... 'tuple'>


smith_form()

The Smith normal form for the Laplacian. In detail: a list of integer matrices $$D, U, V$$ such that $$ULV = D$$ where $$L$$ is the transpose of the Laplacian, $$D$$ is diagonal, and $$U$$ and $$V$$ are invertible over the integers.

OUTPUT:

list of integer matrices

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D,U,V = s.smith_form()
sage: D
[1 0 0 0]
[0 4 0 0]
[0 0 4 0]
[0 0 0 0]
sage: U*s.laplacian()*V == D  # laplacian symmetric => transpose not necessary
True


solve()

Approximations of the complex affine zeros of the sandpile ideal.

OUTPUT:

list of complex numbers

EXAMPLES:

sage: S = Sandpile({0: {}, 1: {2: 2}, 2: {0: 4, 1: 1}}, 0)
sage: S.solve()
[[-0.707107 + 0.707107*I, 0.707107 - 0.707107*I], [-0.707107 - 0.707107*I, 0.707107 + 0.707107*I], [-I, -I], [I, I], [0.707107 + 0.707107*I, -0.707107 - 0.707107*I], [0.707107 - 0.707107*I, -0.707107 + 0.707107*I], [1, 1], [-1, -1]]
sage: len(_)
8
sage: S.group_order()
8


Note

The solutions form a multiplicative group isomorphic to the sandpile group. Generators for this group are given exactly by points().

stable_configs(smax=None)

Generator for all stable configurations. If smax is provided, then the generator gives all stable configurations less than or equal to smax. If smax does not represent a stable configuration, then each component of smax is replaced by the corresponding component of the maximal stable configuration.

INPUT:

smax – (optional) SandpileConfig or list representing a SandpileConfig

OUTPUT:

generator for all stable configurations

EXAMPLES:

sage: s = sandpiles.Complete(3)
sage: a = s.stable_configs()
sage: next(a)
{1: 0, 2: 0}
sage: [i.values() for i in a]
[[0, 1], [1, 0], [1, 1]]
sage: b = s.stable_configs([1,0])
sage: list(b)
[{1: 0, 2: 0}, {1: 1, 2: 0}]


stationary_density()

The stationary density of the sandpile.

OUTPUT:

rational number

EXAMPLES:

sage: s = sandpiles.Complete(3)
sage: s.stationary_density()
10/9
sage: s = Sandpile(digraphs.DeBruijn(2,2),'00')
sage: s.stationary_density()
9/8


Note

The stationary density of a sandpile is the sum $$\sum_c (\deg(c) + \deg(s))$$ where $$\deg(s)$$ is the degree of the sink and the sum is over all recurrent configurations.

REFERENCES:

superstables(verbose=True)

The superstable configurations. If verbose is False, the configurations are converted to lists of integers. Superstables for undirected graphs are also known as G-parking functions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

list of SandpileConfig

EXAMPLES:

sage: sp = Sandpile(graphs.HouseXGraph(),0).superstables()
sage: sp[:3]
[{1: 0, 2: 0, 3: 0, 4: 0}, {1: 1, 2: 0, 3: 0, 4: 1}, {1: 1, 2: 0, 3: 0, 4: 0}]
sage: sandpiles.Complete(4).superstables(False)
[[0, 0, 0],
[0, 0, 1],
[0, 1, 0],
[1, 0, 0],
[0, 0, 2],
[0, 2, 0],
[2, 0, 0],
[0, 1, 1],
[1, 0, 1],
[1, 1, 0],
[0, 1, 2],
[0, 2, 1],
[1, 0, 2],
[1, 2, 0],
[2, 0, 1],
[2, 1, 0]]
sage: sandpiles.Cycle(4).superstables(False)
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]]


symmetric_recurrents(orbits)

The symmetric recurrent configurations.

INPUT:

orbits - list of lists partitioning the vertices

OUTPUT:

list of recurrent configurations

EXAMPLES:

sage: S = Sandpile({0: {},
....:              1: {0: 1, 2: 1, 3: 1},
....:              2: {1: 1, 3: 1, 4: 1},
....:              3: {1: 1, 2: 1, 4: 1},
....:              4: {2: 1, 3: 1}})
sage: S.symmetric_recurrents([[1],[2,3],[4]])
[{1: 2, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 2, 4: 0}]
sage: S.recurrents()
[{1: 2, 2: 2, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 2, 4: 0},
{1: 2, 2: 1, 3: 2, 4: 0},
{1: 2, 2: 2, 3: 0, 4: 1},
{1: 2, 2: 0, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 1, 4: 0},
{1: 2, 2: 1, 3: 2, 4: 1},
{1: 2, 2: 2, 3: 1, 4: 1}]


Note

The user is responsible for ensuring that the list of orbits comes from a group of symmetries of the underlying graph.

tutte_polynomial()

The Tutte polynomial of the underlying graph. Only defined for undirected sandpile graphs.

OUTPUT:

polynomial

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: s.tutte_polynomial()
x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y
sage: s.tutte_polynomial().subs(x=1)
y^3 + 3*y^2 + 6*y + 6
sage: s.tutte_polynomial().subs(x=1).coefficients() == s.h_vector()
True


unsaturated_ideal()

The unsaturated, homogeneous toppling ideal.

OUTPUT:

ideal

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.unsaturated_ideal().gens()
[x1^3 - x3*x2*x0, x2^3 - x3*x1*x0, x3^2 - x2*x1]
sage: S.ideal().gens()
[x2*x1 - x0^2, x3^2 - x0^2, x1^3 - x3*x2*x0, x3*x1^2 - x2^2*x0, x2^3 - x3*x1*x0, x3*x2^2 - x1^2*x0]


version()

The version number of Sage Sandpiles.

OUTPUT:

string

EXAMPLES:

sage: Sandpile.version()
Sage Sandpiles Version 2.4
sage: S = sandpiles.Complete(3)
sage: S.version()
Sage Sandpiles Version 2.4


zero_config()

The all-zero configuration.

OUTPUT:

SandpileConfig

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: s.zero_config()
{1: 0, 2: 0, 3: 0}


zero_div()

The all-zero divisor.

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.House()
sage: S.zero_div()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0}


SandpileConfig¶

Summary of methods.

• + — Addition of configurations.
• & — The stabilization of the sum.
• greater-equalTrue if every component of self is at least that of other.
• greaterTrue if every component of self is at least that of other and the two configurations are not equal.
• ~ — The stabilized configuration.
• less-equalTrue if every component of self is at most that of other.
• lessTrue if every component of self is at most that of other and the two configurations are not equal.
• * — The recurrent element equivalent to the sum.
• ^ — Exponentiation for the *-operator.
• - — The additive inverse of the configuration.
• - — Subtraction of configurations.
• add_random — Add one grain of sand to a random vertex.
• burst_size — The burst size of the configuration with respect to the given vertex.
• deg — The degree of the configuration.
• dualize — The difference with the maximal stable configuration.
• equivalent_recurrent — The recurrent configuration equivalent to the given configuration.
• equivalent_superstable — The equivalent superstable configuration.
• fire_script — Fire the given script.
• fire_unstable — Fire all unstable vertices.
• fire_vertex — Fire the given vertex.
• help — List of SandpileConfig methods.
• is_recurrent — Is the configuration recurrent?
• is_stable — Is the configuration stable?
• is_superstable — Is the configuration superstable?
• is_symmetric — Is the configuration symmetric?
• order — The order of the equivalent recurrent element.
• sandpile — The configuration’s underlying sandpile.
• show — Show the configuration.
• stabilize — The stabilized configuration.
• support — The vertices containing sand.
• unstable — The unstable vertices.
• values — The values of the configuration as a list.

Complete descriptions of SandpileConfig methods.

+

INPUT:

other – SandpileConfig

OUTPUT:

sum of self and other

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [3,2])
sage: c + d
{1: 4, 2: 4}


&

The stabilization of the sum.

INPUT:

other – SandpileConfig

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,0,0])
sage: c + c  # ordinary addition
{1: 2, 2: 0, 3: 0}
sage: c & c  # add and stabilize
{1: 0, 2: 1, 3: 0}
sage: c*c  # add and find equivalent recurrent
{1: 1, 2: 1, 3: 1}
sage: ~(c + c) == c & c
True


>=

True if every component of self is at least that of other.

INPUT:

other – SandpileConfig

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [2,3])
sage: e = SandpileConfig(S, [2,0])
sage: c >= c
True
sage: d >= c
True
sage: c >= d
False
sage: e >= c
False
sage: c >= e
False


>

True if every component of self is at least that of other and the two configurations are not equal.

INPUT:

other – SandpileConfig

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [1,3])
sage: c > c
False
sage: d > c
True
sage: c > d
False


~

The stabilized configuration.

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.House()
sage: c = S.max_stable() + S.identity()
sage: ~c == c.stabilize()
True


<=

True if every component of self is at most that of other.

INPUT:

other – SandpileConfig

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [2,3])
sage: e = SandpileConfig(S, [2,0])
sage: c <= c
True
sage: c <= d
True
sage: d <= c
False
sage: c <= e
False
sage: e <= c
False


<

True if every component of self is at most that of other and the two configurations are not equal.

INPUT:

other – SandpileConfig

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [2,3])
sage: c < c
False
sage: c < d
True
sage: d < c
False
sage: S = Sandpile(graphs.CycleGraph(3), 0)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [2,3])
sage: c < c
False
sage: c < d
True
sage: d < c
False


*

If other is an configuration, the recurrent element equivalent to the sum. If other is an integer, the sum of configuration with itself other times.

INPUT:

other – SandpileConfig or Integer

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,0,0])
sage: c + c  # ordinary addition
{1: 2, 2: 0, 3: 0}
sage: c & c  # add and stabilize
{1: 0, 2: 1, 3: 0}
sage: c*c  # add and find equivalent recurrent
{1: 1, 2: 1, 3: 1}
sage: (c*c).is_recurrent()
True
sage: c*(-c) == S.identity()
True
sage: c
{1: 1, 2: 0, 3: 0}
sage: c*3
{1: 3, 2: 0, 3: 0}


^

The recurrent element equivalent to the sum of the configuration with itself $$k$$ times. If $$k$$ is negative, do the same for the negation of the configuration. If $$k$$ is zero, return the identity of the sandpile group.

INPUT:

k – SandpileConfig

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,0,0])
sage: c^3
{1: 1, 2: 1, 3: 0}
sage: (c + c + c) == c^3
False
sage: (c + c + c).equivalent_recurrent() == c^3
True
sage: c^(-1)
{1: 1, 2: 1, 3: 0}
sage: c^0 == S.identity()
True


_

The additive inverse of the configuration.

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: -c
{1: -1, 2: -2}


-

Subtraction of configurations.

INPUT:

other – SandpileConfig

OUTPUT:

sum of self and other

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: d = SandpileConfig(S, [3,2])
sage: c - d
{1: -2, 2: 0}


Add one grain of sand to a random vertex. Optionally, a probability distribution, distrib, may be placed on the vertices or the nonsink vertices. See NOTE for details.

INPUT:

distrib – (optional) list of nonnegative numbers summing to 1 (representing a prob. dist.)

OUTPUT:

SandpileConfig

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: c = s.zero_config()
{1: 0, 2: 1, 3: 0}
sage: c
{1: 0, 2: 0, 3: 0}
{1: 0, 2: 0, 3: 1}
{1: 0, 2: 0, 3: 0}


We compute the “sizes” of the avalanches caused by adding random grains of sand to the maximal stable configuration on a grid graph. The function stabilize() returns the firing vector of the stabilization, a dictionary whose values say how many times each vertex fires in the stabilization.:

sage: S = sandpiles.Grid(10,10)
sage: m = S.max_stable()
sage: a = []
sage: for i in range(1000):
....:     m, f = m.stabilize(True)
....:     a.append(sum(f.values()))
sage: p = list_plot([[log(i+1),log(a.count(i))] for i in [0..max(a)] if a.count(i)])
sage: p.axes_labels(['log(N)','log(D(N))'])
sage: t = text("Distribution of avalanche sizes", (2,2), rgbcolor=(1,0,0))
sage: show(p+t,axes_labels=['log(N)','log(D(N))'])


Note

If distrib is None, then the probability is the uniform probability on the nonsink vertices. Otherwise, there are two possibilities:

(i) the length of distrib is equal to the number of vertices, and distrib represents a probability distribution on all of the vertices. In that case, the sink may be chosen at random, in which case, the configuration is unchanged.

(ii) Otherwise, the length of distrib must be equal to the number of nonsink vertices, and distrib represents a probability distribution on the nonsink vertices.

Warning

If distrib != None, the user is responsible for assuring the sum of its entries is 1 and that its length is equal to the number of sink vertices or the number of nonsink vertices.

burst_size(v)

The burst size of the configuration with respect to the given vertex.

INPUT:

v – vertex

OUTPUT:

integer

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: [i.burst_size(0) for i in s.recurrents()]
[1, 1, 1, 1, 1, 1, 1, 1]
sage: [i.burst_size(1) for i in s.recurrents()]
[0, 0, 1, 2, 1, 2, 0, 2]


Note

To define c.burst(v), if $$v$$ is not the sink, let $$c'$$ be the unique recurrent for which the stabilization of $$c' + v$$ is $$c$$. The burst size is then the amount of sand that goes into the sink during this stabilization. If $$v$$ is the sink, the burst size is defined to be 1.

REFERENCES:

deg()

The degree of the configuration.

OUTPUT:

integer

EXAMPLES:

sage: S = sandpiles.Complete(3)
sage: c = SandpileConfig(S, [1,2])
sage: c.deg()
3


dualize()

The difference with the maximal stable configuration.

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: S.max_stable()
{1: 1, 2: 1}
sage: c.dualize()
{1: 0, 2: -1}
sage: S.max_stable() - c == c.dualize()
True


equivalent_recurrent(with_firing_vector=False)

The recurrent configuration equivalent to the given configuration. Optionally, return the corresponding firing vector.

INPUT:

with_firing_vector – (default: False) boolean

OUTPUT:

SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: c = SandpileConfig(S, [0,0,0])
sage: c.equivalent_recurrent() == S.identity()
True
sage: x = c.equivalent_recurrent(True)
sage: r = vector([x[0][v] for v in S.nonsink_vertices()])
sage: f = vector([x[1][v] for v in S.nonsink_vertices()])
sage: cv = vector(c.values())
sage: r == cv - f*S.reduced_laplacian()
True


Note

Let $$L$$ be the reduced Laplacian, $$c$$ the initial configuration, $$r$$ the returned configuration, and $$f$$ the firing vector. Then $$r = c - f\cdot L$$.

equivalent_superstable(with_firing_vector=False)

The equivalent superstable configuration. Optionally, return the corresponding firing vector.

INPUT:

with_firing_vector – (default: False) boolean

OUTPUT:

SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: m = S.max_stable()
sage: m.equivalent_superstable().is_superstable()
True
sage: x = m.equivalent_superstable(True)
sage: s = vector(x[0].values())
sage: f = vector(x[1].values())
sage: mv = vector(m.values())
sage: s == mv - f*S.reduced_laplacian()
True


Note

Let $$L$$ be the reduced Laplacian, $$c$$ the initial configuration, $$s$$ the returned configuration, and $$f$$ the firing vector. Then $$s = c - f\cdot L$$.

fire_script(sigma)

Fire the given script. In other words, fire each vertex the number of times indicated by sigma.

INPUT:

sigma – SandpileConfig or (list or dict representing a SandpileConfig)

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.unstable()
[2, 3]
sage: c.fire_script(SandpileConfig(S,[0,1,1]))
{1: 2, 2: 1, 3: 2}
sage: c.fire_script(SandpileConfig(S,[2,0,0])) == c.fire_vertex(1).fire_vertex(1)
True


fire_unstable()

Fire all unstable vertices.

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.fire_unstable()
{1: 2, 2: 1, 3: 2}


fire_vertex(v)

Fire the given vertex.

INPUT:

v – vertex

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: c = SandpileConfig(S, [1,2])
sage: c.fire_vertex(2)
{1: 2, 2: 0}


help(verbose=True)

List of SandpileConfig methods. If verbose, include short descriptions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

printed string

EXAMPLES:

sage: SandpileConfig.help()
Shortcuts for SandpileConfig operations:
~c    -- stabilize
c & d -- add and stabilize
c * c -- add and find equivalent recurrent
c^k   -- add k times and find equivalent recurrent
(taking inverse if k is negative)

For detailed help with any method FOO listed below,
enter "SandpileConfig.FOO?" or enter "c.FOO?" for any SandpileConfig c.

burst_size             -- The burst size of the configuration with respect to the given vertex.
deg                    -- The degree of the configuration.
dualize                -- The difference with the maximal stable configuration.
equivalent_recurrent   -- The recurrent configuration equivalent to the given configuration.
equivalent_superstable -- The equivalent superstable configuration.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileConfig methods.
is_recurrent           -- Is the configuration recurrent?
is_stable              -- Is the configuration stable?
is_superstable         -- Is the configuration superstable?
is_symmetric           -- Is the configuration symmetric?
order                  -- The order of the equivalent recurrent element.
sandpile               -- The configuration's underlying sandpile.
show                   -- Show the configuration.
stabilize              -- The stabilized configuration.
support                -- The vertices containing sand.
unstable               -- The unstable vertices.
values                 -- The values of the configuration as a list.


is_recurrent()

Is the configuration recurrent?

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.identity().is_recurrent()
True
sage: S.zero_config().is_recurrent()
False


is_stable()

Is the configuration stable?

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.max_stable().is_stable()
True
sage: (2*S.max_stable()).is_stable()
False
sage: (S.max_stable() & S.max_stable()).is_stable()
True


is_superstable()

Is the configuration superstable?

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: S.zero_config().is_superstable()
True


is_symmetric(orbits)

Is the configuration symmetric? Return True if the values of the configuration are constant over the vertices in each sublist of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT:

boolean

EXAMPLES:

sage: S = Sandpile({0: {},
....:              1: {0: 1, 2: 1, 3: 1},
....:              2: {1: 1, 3: 1, 4: 1},
....:              3: {1: 1, 2: 1, 4: 1},
....:              4: {2: 1, 3: 1}})
sage: c = SandpileConfig(S, [1, 2, 2, 3])
sage: c.is_symmetric([[2,3]])
True


order()

The order of the equivalent recurrent element.

OUTPUT:

integer

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: c = SandpileConfig(S,[2,0,1])
sage: c.order()
4
sage: ~(c + c + c + c) == S.identity()
True
sage: c = SandpileConfig(S,[1,1,0])
sage: c.order()
1
sage: c.is_recurrent()
False
sage: c.equivalent_recurrent() == S.identity()
True


sandpile()

The configuration’s underlying sandpile.

OUTPUT:

Sandpile

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
sage: c.sandpile() == S
True


show(sink=True, colors=True, heights=False, directed=None, **kwds)

Show the configuration.

INPUT:

• sink – (default: True) whether to show the sink
• colors – (default: True) whether to color-code the amount of sand on each vertex
• heights – (default: False) whether to label each vertex with the amount of sand
• directed – (optional) whether to draw directed edges
• kwds – (optional) arguments passed to the show method for Graph

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c.show()
sage: c.show(directed=False)
sage: c.show(sink=False,colors=False,heights=True)


stabilize(with_firing_vector=False)

The stabilized configuration. Optionally returns the corresponding firing vector.

INPUT:

with_firing_vector – (default: False) boolean

OUTPUT:

SandpileConfig or [SandpileConfig, firing_vector]

EXAMPLES:

sage: S = sandpiles.House()
sage: c = 2*S.max_stable()
sage: c._set_stabilize()
sage: '_stabilize' in c.__dict__
True
sage: S = sandpiles.House()
sage: c = S.max_stable() + S.identity()
sage: c.stabilize(True)
[{1: 1, 2: 2, 3: 2, 4: 1}, {1: 2, 2: 2, 3: 3, 4: 3}]
sage: S.max_stable() & S.identity() == c.stabilize()
True
sage: ~c == c.stabilize()
True


support()

The vertices containing sand.

OUTPUT:

list - support of the configuration

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: c = S.identity()
sage: c
{1: 2, 2: 2, 3: 0}
sage: c.support()
[1, 2]


unstable()

The unstable vertices.

OUTPUT:

list of vertices

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: c = SandpileConfig(S, [1,2,3])
sage: c.unstable()
[2, 3]


values()

The values of the configuration as a list. The list is sorted in the order of the vertices.

OUTPUT:

list of integers

boolean

EXAMPLES:

sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
sage: c = SandpileConfig(S, {'b':1, 'c':2})
sage: c
{'b': 1, 'c': 2}
sage: c.values()
[1, 2]
sage: S.nonsink_vertices()
['b', 'c']


SandpileDivisor¶

Summary of methods.

• + — Addition of divisors.
• greater-equalTrue if every component of self is at least that of other.
• greaterTrue if every component of self is at least that of other and the two divisors are not equal.
• less-equalTrue if every component of self is at most that of other.
• lessTrue if every component of self is at most that of other and the two divisors are not equal.
• - — The additive inverse of the divisor.
• - — Subtraction of divisors.
• Dcomplex — The support-complex.
• add_random — Add one grain of sand to a random vertex.
• betti — The Betti numbers for the support-complex.
• deg — The degree of the divisor.
• dualize — The difference with the maximal stable divisor.
• effective_div — All linearly equivalent effective divisors.
• fire_script — Fire the given script.
• fire_unstable — Fire all unstable vertices.
• fire_vertex — Fire the given vertex.
• help — List of SandpileDivisor methods.
• is_alive — Is the divisor stabilizable?
• is_linearly_equivalent — Is the given divisor linearly equivalent?
• is_q_reduced — Is the divisor q-reduced?
• is_symmetric — Is the divisor symmetric?
• is_weierstrass_pt — Is the given vertex a Weierstrass point?
• polytope — The polytope determining the complete linear system.
• polytope_integer_pts — The integer points inside divisor’s polytope.
• q_reduced — The linearly equivalent q-reduced divisor.
• rank — The rank of the divisor.
• sandpile — The divisor’s underlying sandpile.
• show — Show the divisor.
• simulate_threshold — The first unstabilizable divisor in the closed Markov chain.
• stabilize — The stabilization of the divisor.
• support — List of vertices at which the divisor is nonzero.
• unstable — The unstable vertices.
• values — The values of the divisor as a list.
• weierstrass_div — The Weierstrass divisor.
• weierstrass_gap_seq — The Weierstrass gap sequence at the given vertex.
• weierstrass_pts — The Weierstrass points (vertices).
• weierstrass_rank_seq — The Weierstrass rank sequence at the given vertex.

Complete descriptions of SandpileDivisor methods.

+

INPUT:

other – SandpileDivisor

OUTPUT:

sum of self and other

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [3,2,1])
sage: D + E
{0: 4, 1: 4, 2: 4}


>=

True if every component of self is at least that of other.

INPUT:

other – SandpileDivisor

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [2,3,4])
sage: F = SandpileDivisor(S, [2,0,4])
sage: D >= D
True
sage: E >= D
True
sage: D >= E
False
sage: F >= D
False
sage: D >= F
False


>

True if every component of self is at least that of other and the two divisors are not equal.

INPUT:

other – SandpileDivisor

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [1,3,4])
sage: D > D
False
sage: E > D
True
sage: D > E
False


<=

True if every component of self is at most that of other.

INPUT:

other – SandpileDivisor

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [2,3,4])
sage: F = SandpileDivisor(S, [2,0,4])
sage: D <= D
True
sage: D <= E
True
sage: E <= D
False
sage: D <= F
False
sage: F <= D
False


<

True if every component of self is at most that of other and the two divisors are not equal.

INPUT:

other – SandpileDivisor

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [2,3,4])
sage: D < D
False
sage: D < E
True
sage: E < D
False


-

The additive inverse of the divisor.

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: -D
{0: -1, 1: -2, 2: -3}


-

Subtraction of divisors.

INPUT:

other – SandpileDivisor

OUTPUT:

Difference of self and other

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: E = SandpileDivisor(S, [3,2,1])
sage: D - E
{0: -2, 1: 0, 2: 2}


Dcomplex()

The support-complex. (See NOTE.)

OUTPUT:

simplicial complex

EXAMPLES:

sage: S = sandpiles.House()
sage: p = SandpileDivisor(S, [1,2,1,0,0]).Dcomplex()
sage: p.homology()
{0: 0, 1: Z x Z, 2: 0}
sage: p.f_vector()
[1, 5, 10, 4]
sage: p.betti()
{0: 1, 1: 2, 2: 0}


Note

The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.

Add one grain of sand to a random vertex.

INPUT:

distrib – (optional) list of nonnegative numbers representing a probability distribution on the vertices

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = s.zero_div()
{0: 0, 1: 0, 2: 1, 3: 0}
{0: 0, 1: 0, 2: 0, 3: 1}


Warning

If distrib is not None, the user is responsible for assuring the sum of its entries is 1.

betti()

The Betti numbers for the support-complex. (See NOTE.)

OUTPUT:

dictionary of integers

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [2,0,1])
sage: D.betti()
{0: 1, 1: 1}


Note

The “support-complex” is the simplicial complex determined by the supports of the linearly equivalent effective divisors.

deg()

The degree of the divisor.

OUTPUT:

integer

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.deg()
6


dualize()

The difference with the maximal stable divisor.

OUTPUT:

SandpileDivisor

EXAMPLES::
sage: S = sandpiles.Cycle(3) sage: D = SandpileDivisor(S, [1,2,3]) sage: D.dualize() {0: 0, 1: -1, 2: -2} sage: S.max_stable_div() - D == D.dualize() True

effective_div(verbose=True, with_firing_vectors=False)

All linearly equivalent effective divisors. If verbose is False, the divisors are converted to lists of integers. If with_firing_vectors is True then a list of firing vectors is also given, each of which prescribes the vertices to be fired in order to obtain an effective divisor.

INPUT:

• verbose – (default: True) boolean
• with_firing_vectors – (default: False) boolean

OUTPUT:

list (of divisors)

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: sorted(D.effective_div(), key=str)
[{0: 0, 1: 2, 2: 0, 3: 4},
{0: 0, 1: 2, 2: 4, 3: 0},
{0: 0, 1: 6, 2: 0, 3: 0},
{0: 1, 1: 3, 2: 1, 3: 1},
{0: 2, 1: 0, 2: 2, 3: 2},
{0: 4, 1: 2, 2: 0, 3: 0}]
sage: sorted(D.effective_div(False))
[[0, 2, 0, 4],
[0, 2, 4, 0],
[0, 6, 0, 0],
[1, 3, 1, 1],
[2, 0, 2, 2],
[4, 2, 0, 0]]
sage: sorted(D.effective_div(with_firing_vectors=True), key=str)
[({0: 0, 1: 2, 2: 0, 3: 4}, (0, -1, -1, -2)),
({0: 0, 1: 2, 2: 4, 3: 0}, (0, -1, -2, -1)),
({0: 0, 1: 6, 2: 0, 3: 0}, (0, -2, -1, -1)),
({0: 1, 1: 3, 2: 1, 3: 1}, (0, -1, -1, -1)),
({0: 2, 1: 0, 2: 2, 3: 2}, (0, 0, -1, -1)),
({0: 4, 1: 2, 2: 0, 3: 0}, (0, 0, 0, 0))]
sage: a = _[2]
sage: a[0].values()
[0, 6, 0, 0]
sage: vector(D.values()) - s.laplacian()*a[1]
(0, 6, 0, 0)
sage: sorted(D.effective_div(False, True))
[([0, 2, 0, 4], (0, -1, -1, -2)),
([0, 2, 4, 0], (0, -1, -2, -1)),
([0, 6, 0, 0], (0, -2, -1, -1)),
([1, 3, 1, 1], (0, -1, -1, -1)),
([2, 0, 2, 2], (0, 0, -1, -1)),
([4, 2, 0, 0], (0, 0, 0, 0))]
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.effective_div(False,True)
[]


fire_script(sigma)

Fire the given script. In other words, fire each vertex the number of times indicated by sigma.

INPUT:

sigma – SandpileDivisor or (list or dict representing a SandpileDivisor)

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.unstable()
[1, 2]
sage: D.fire_script([0,1,1])
{0: 3, 1: 1, 2: 2}
sage: D.fire_script(SandpileDivisor(S,[2,0,0])) == D.fire_vertex(0).fire_vertex(0)
True


fire_unstable()

Fire all unstable vertices.

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.fire_unstable()
{0: 3, 1: 1, 2: 2}


fire_vertex(v)

Fire the given vertex.

INPUT:

v – vertex

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.fire_vertex(1)
{0: 2, 1: 0, 2: 4}


help(verbose=True)

List of SandpileDivisor methods. If verbose, include short descriptions.

INPUT:

verbose – (default: True) boolean

OUTPUT:

printed string

EXAMPLES:

sage: SandpileDivisor.help()
For detailed help with any method FOO listed below,
enter "SandpileDivisor.FOO?" or enter "D.FOO?" for any SandpileDivisor D.

Dcomplex               -- The support-complex.
betti                  -- The Betti numbers for the support-complex.
deg                    -- The degree of the divisor.
dualize                -- The difference with the maximal stable divisor.
effective_div          -- All linearly equivalent effective divisors.
fire_script            -- Fire the given script.
fire_unstable          -- Fire all unstable vertices.
fire_vertex            -- Fire the given vertex.
help                   -- List of SandpileDivisor methods.
is_alive               -- Is the divisor stabilizable?
is_linearly_equivalent -- Is the given divisor linearly equivalent?
is_q_reduced           -- Is the divisor q-reduced?
is_symmetric           -- Is the divisor symmetric?
is_weierstrass_pt      -- Is the given vertex a Weierstrass point?
polytope               -- The polytope determining the complete linear system.
polytope_integer_pts   -- The integer points inside divisor's polytope.
q_reduced              -- The linearly equivalent q-reduced divisor.
rank                   -- The rank of the divisor.
sandpile               -- The divisor's underlying sandpile.
show                   -- Show the divisor.
simulate_threshold     -- The first unstabilizable divisor in the closed Markov chain.
stabilize              -- The stabilization of the divisor.
support                -- List of vertices at which the divisor is nonzero.
unstable               -- The unstable vertices.
values                 -- The values of the divisor as a list.
weierstrass_div        -- The Weierstrass divisor.
weierstrass_gap_seq    -- The Weierstrass gap sequence at the given vertex.
weierstrass_pts        -- The Weierstrass points (vertices).
weierstrass_rank_seq   -- The Weierstrass rank sequence at the given vertex.


is_alive(cycle=False)

Is the divisor stabilizable? In other words, will the divisor stabilize under repeated firings of all unstable vertices? Optionally returns the resulting cycle.

INPUT:

cycle – (default: False) boolean

OUTPUT:

boolean or optionally, a list of SandpileDivisors

EXAMPLES:

sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S, {0: 4, 1: 3, 2: 3, 3: 2})
sage: D.is_alive()
True
sage: D.is_alive(True)
[{0: 4, 1: 3, 2: 3, 3: 2}, {0: 3, 1: 2, 2: 2, 3: 5}, {0: 1, 1: 4, 2: 4, 3: 3}]


is_linearly_equivalent(D, with_firing_vector=False)

Is the given divisor linearly equivalent? Optionally, returns the firing vector. (See NOTE.)

INPUT:

• D – SandpileDivisor or list, tuple, etc. representing a divisor
• with_firing_vector – (default: False) boolean

OUTPUT:

boolean or integer vector

EXAMPLES:

sage: s = sandpiles.Complete(3)
sage: D = SandpileDivisor(s,[2,0,0])
sage: D.is_linearly_equivalent([0,1,1])
True
sage: D.is_linearly_equivalent([0,1,1],True)
(1, 0, 0)
sage: v = vector(D.is_linearly_equivalent([0,1,1],True))
sage: vector(D.values()) - s.laplacian()*v
(0, 1, 1)
sage: D.is_linearly_equivalent([0,0,0])
False
sage: D.is_linearly_equivalent([0,0,0],True)
()


Note

• If with_firing_vector is False, returns either True or False.
• If with_firing_vector is True then: (i) if self is linearly equivalent to $$D$$, returns a vector $$v$$ such that self - v*self.laplacian().transpose() = D. Otherwise, (ii) if self is not linearly equivalent to $$D$$, the output is the empty vector, ().

is_q_reduced()

Is the divisor $$q$$-reduced? This would mean that $$self = c + kq$$ where $$c$$ is superstable, $$k$$ is an integer, and $$q$$ is the sink vertex.

OUTPUT:

boolean

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[2,-3,2,0])
sage: D.is_q_reduced()
False
sage: SandpileDivisor(s,[10,0,1,2]).is_q_reduced()
True


For undirected or, more generally, Eulerian graphs, $$q$$-reduced divisors are linearly equivalent if and only if they are equal. The same does not hold for general directed graphs:

sage: s = Sandpile({0:[1],1:[1,1]})
sage: D = SandpileDivisor(s,[-1,1])
sage: Z = s.zero_div()
sage: D.is_q_reduced()
True
sage: Z.is_q_reduced()
True
sage: D == Z
False
sage: D.is_linearly_equivalent(Z)
True


is_symmetric(orbits)

Is the divisor symmetric? Return True if the values of the configuration are constant over the vertices in each sublist of orbits.

INPUT:

orbits – list of lists of vertices

OUTPUT:

boolean

EXAMPLES:

sage: S = sandpiles.House()
sage: S.dict()
{0: {1: 1, 2: 1},
1: {0: 1, 3: 1},
2: {0: 1, 3: 1, 4: 1},
3: {1: 1, 2: 1, 4: 1},
4: {2: 1, 3: 1}}
sage: D = SandpileDivisor(S, [0,0,1,1,3])
sage: D.is_symmetric([[2,3], [4]])
True


is_weierstrass_pt(v=’sink’)

Is the given vertex a Weierstrass point?

INPUT:

v – (default: sink) vertex

OUTPUT:

boolean

EXAMPLES:

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: K.weierstrass_rank_seq()  # sequence at the sink vertex, 0
(1, 0, -1)
sage: K.is_weierstrass_pt()
False
sage: K.weierstrass_rank_seq(4)
(1, 0, 0, -1)
sage: K.is_weierstrass_pt(4)
True


Note

The vertex $$v$$ is a (generalized) Weierstrass point for divisor $$D$$ if the sequence of ranks $$r(D - nv)$$ for $$n = 0, 1, 2, \dots$$ is not $$r(D), r(D)-1, \dots, 0, -1, -1, \dots$$.

polytope()

The polytope determining the complete linear system.

OUTPUT:

polytope

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: p = D.polytope()
sage: p.inequalities()
(An inequality (-3, 1, 1) x + 2 >= 0,
An inequality (1, 1, 1) x + 4 >= 0,
An inequality (1, -3, 1) x + 0 >= 0,
An inequality (1, 1, -3) x + 0 >= 0)
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.polytope()
The empty polyhedron in QQ^3


Note

For a divisor $$D$$, this is the intersection of (i) the polyhedron determined by the system of inequalities $$L^t x \leq D$$ where $$L^t$$ is the transpose of the Laplacian with (ii) the hyperplane $$x_{\mathrm{sink\_vertex}} = 0$$. The polytope is thought of as sitting in $$(n-1)$$-dimensional Euclidean space where $$n$$ is the number of vertices.

polytope_integer_pts()

The integer points inside divisor’s polytope. The polytope referred to here is the one determining the divisor’s complete linear system (see the documentation for polytope).

OUTPUT:

tuple of integer vectors

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[4,2,0,0])
sage: sorted(D.polytope_integer_pts())
[(-2, -1, -1),
(-1, -2, -1),
(-1, -1, -2),
(-1, -1, -1),
(0, -1, -1),
(0, 0, 0)]
sage: D = SandpileDivisor(s,[-1,0,0,0])
sage: D.polytope_integer_pts()
()


q_reduced(verbose=True)

The linearly equivalent $$q$$-reduced divisor.

INPUT:

verbose – (default: True) boolean

OUTPUT:

SandpileDivisor or list representing SandpileDivisor

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[2,-3,2,0])
sage: D.q_reduced()
{0: -2, 1: 1, 2: 2, 3: 0}
sage: D.q_reduced(False)
[-2, 1, 2, 0]


Note

The divisor $$D$$ is $$qreduced if$$ where $$c$$ is superstable, $$k$$ is an integer, and $$q$$ is the sink.

rank(with_witness=False)

The rank of the divisor. Optionally returns an effective divisor $$E$$ such that $$D - E$$ is not winnable (has an empty complete linear system).

INPUT:

with_witness – (default: False) boolean

OUTPUT:

integer or (integer, SandpileDivisor)

EXAMPLES:

   sage: S = sandpiles.Complete(4)
sage: D = SandpileDivisor(S,[4,2,0,0])
sage: D.rank()
3
sage: D.rank(True)
(3, {0: 3, 1: 0, 2: 1, 3: 0})
sage: E = _[1]
sage: (D - E).rank()
-1

Riemann-Roch theorem::

sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus()
True

Riemann-Roch theorem::

sage: D.rank() - (S.canonical_divisor()-D).rank() == D.deg() + 1 - S.genus()
True
sage: S = Sandpile({0:[1,1,1,2],1:[0,0,0,1,1,1,2,2],2:[2,2,1,1,0]},0) # multigraph with loops
sage: D = SandpileDivisor(S,[4,2,0])
sage: D.rank(True)
(2, {0: 1, 1: 1, 2: 1})
sage: S = Sandpile({0:[1,2], 1:[0,2,2], 2: [0,1]},0) # directed graph
sage: S.is_undirected()
False
sage: D = SandpileDivisor(S,[0,2,0])
sage: D.effective_div()
[{0: 0, 1: 2, 2: 0}, {0: 2, 1: 0, 2: 0}]
sage: D.rank(True)
(0, {0: 0, 1: 0, 2: 1})
sage: E = D.rank(True)[1]
sage: (D - E).effective_div()
[]


Note

The rank of a divisor $$D$$ is -1 if $$D$$ is not linearly equivalent to an effective divisor (i.e., the dollar game represented by $$D$$ is unwinnable). Otherwise, the rank of $$D$$ is the largest integer $$r$$ such that $$D - E$$ is linearly equivalent to an effective divisor for all effective divisors $$E$$ with $$\deg(E) = r$$.

sandpile()

The divisor’s underlying sandpile.

OUTPUT:

Sandpile

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: D = SandpileDivisor(S,[1,-2,0,3])
sage: D.sandpile()
Diamond sandpile graph: 4 vertices, sink = 0
sage: D.sandpile() == S
True


show(heights=True, directed=None, **kwds)

Show the divisor.

INPUT:

• heights – (default: True) whether to label each vertex with the amount of sand
• directed – (optional) whether to draw directed edges
• kwds – (optional) arguments passed to the show method for Graph

EXAMPLES:

sage: S = sandpiles.Diamond()
sage: D = SandpileDivisor(S,[1,-2,0,2])
sage: D.show(graph_border=True,vertex_size=700,directed=False)


simulate_threshold(distrib=None)

The first unstabilizable divisor in the closed Markov chain. (See NOTE.)

INPUT:

distrib – (optional) list of nonnegative numbers representing a probability distribution on the vertices

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = s.zero_div()
sage: D.simulate_threshold()  # random
{0: 2, 1: 3, 2: 1, 3: 2}
sage: n(mean([D.simulate_threshold().deg() for _ in range(10)]))  # random
7.10000000000000
sage: n(s.stationary_density()*s.num_verts())
6.93750000000000


Note

Starting at self, repeatedly choose a vertex and add a grain of sand to it. Return the first unstabilizable divisor that is reached. Also see the markov_chain method for the underlying sandpile.

stabilize(with_firing_vector=False)

The stabilization of the divisor. If not stabilizable, return an error.

INPUT:

with_firing_vector – (default: False) boolean

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: D = SandpileDivisor(s,[0,3,0,0])
sage: D.stabilize()
{0: 1, 1: 0, 2: 1, 3: 1}
sage: D.stabilize(with_firing_vector=True)
[{0: 1, 1: 0, 2: 1, 3: 1}, {0: 0, 1: 1, 2: 0, 3: 0}]


support()

List of vertices at which the divisor is nonzero.

OUTPUT:

list representing the support of the divisor

EXAMPLES:

sage: S = sandpiles.Cycle(4)
sage: D = SandpileDivisor(S, [0,0,1,1])
sage: D.support()
[2, 3]
sage: S.vertices()
[0, 1, 2, 3]


unstable()

The unstable vertices.

OUTPUT:

list of vertices

EXAMPLES:

sage: S = sandpiles.Cycle(3)
sage: D = SandpileDivisor(S, [1,2,3])
sage: D.unstable()
[1, 2]


values()

The values of the divisor as a list. The list is sorted in the order of the vertices.

OUTPUT:

list of integers

boolean

EXAMPLES:

sage: S = Sandpile({'a':['c','b'], 'b':['c','a'], 'c':['a']},'a')
sage: D = SandpileDivisor(S, {'a':0, 'b':1, 'c':2})
sage: D
{'a': 0, 'b': 1, 'c': 2}
sage: D.values()
[0, 1, 2]
sage: S.vertices()
['a', 'b', 'c']


weierstrass_div(verbose=True)

The Weierstrass divisor. Its value at a vertex is the weight of that vertex as a Weierstrass point. (See SandpileDivisor.weierstrass_gap_seq.)

INPUT:

verbose – (default: True) boolean

OUTPUT:

SandpileDivisor

EXAMPLES:

sage: s = sandpiles.Diamond()
sage: D = SandpileDivisor(s,[4,2,1,0])
sage: [D.weierstrass_rank_seq(v) for v in s]
[(5, 4, 3, 2, 1, 0, 0, -1),
(5, 4, 3, 2, 1, 0, -1),
(5, 4, 3, 2, 1, 0, 0, 0, -1),
(5, 4, 3, 2, 1, 0, 0, -1)]
sage: D.weierstrass_div()
{0: 1, 1: 0, 2: 2, 3: 1}
sage: k5 = sandpiles.Complete(5)
sage: K = k5.canonical_divisor()
sage: K.weierstrass_div()
{0: 9, 1: 9, 2: 9, 3: 9, 4: 9}


weierstrass_gap_seq(v=’sink’, weight=True)

The Weierstrass gap sequence at the given vertex. If weight is True, then also compute the weight of each gap value.

INPUT:

• v – (default: sink) vertex
• weight – (default: True) boolean

OUTPUT:

list or (list of list) of integers

EXAMPLES:

sage: s = sandpiles.Cycle(4)
sage: D = SandpileDivisor(s,[2,0,0,0])
sage: [D.weierstrass_gap_seq(v,False) for v in s.vertices()]
[(1, 3), (1, 2), (1, 3), (1, 2)]
sage: [D.weierstrass_gap_seq(v) for v in s.vertices()]
[((1, 3), 1), ((1, 2), 0), ((1, 3), 1), ((1, 2), 0)]
sage: D.weierstrass_gap_seq()  # gap sequence at sink vertex, 0
((1, 3), 1)
sage: D.weierstrass_rank_seq()  # rank sequence at the sink vertex
(1, 0, 0, -1)


Note

The integer $$k$$ is a Weierstrass gap for the divisor $$D$$ at vertex $$v$$ if the rank of $$D - (k-1)v$$ does not equal the rank of $$D - kv$$. Let $$r$$ be the rank of $$D$$ and let $$k_i$$ be the $$i$$-th gap at $$v$$. The Weierstrass weight of $$v$$ for $$D$$ is the sum of $$(k_i - i)$$ as $$i$$ ranges from $$1$$ to $$r + 1$$. It measure the difference between the sequence $$r, r - 1, ..., 0, -1, -1, ...$$ and the rank sequence $$\mathrm{rank}(D), \mathrm{rank}(D - v), \mathrm{rank}(D - 2v), \dots$$

weierstrass_pts(with_rank_seq=False)

The Weierstrass points (vertices). Optionally, return the corresponding rank sequences.

INPUT:

with_rank_seq – (default: False) boolean

OUTPUT:

tuple of vertices or list of (vertex, rank sequence)

EXAMPLES:

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: K.weierstrass_pts()
(4,)
sage: K.weierstrass_pts(True)
[(4, (1, 0, 0, -1))]


Note

The vertex $$v$$ is a (generalized) Weierstrass point for divisor $$D$$ if the sequence of ranks $$r(D - nv)$$ for $$n = 0, 1, 2, \dots$$ is not $$r(D), r(D)-1, \dots, 0, -1, -1, \dots$$

weierstrass_rank_seq(v=’sink’)

The Weierstrass rank sequence at the given vertex. Computes the rank of the divisor $$D - nv$$ starting with $$n=0$$ and ending when the rank is $$-1$$.

INPUT:

v – (default: sink) vertex

OUTPUT:

tuple of int

EXAMPLES:

sage: s = sandpiles.House()
sage: K = s.canonical_divisor()
sage: [K.weierstrass_rank_seq(v) for v in s.vertices()]
[(1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, -1), (1, 0, 0, -1)]


Other¶

Complete descriptions of methods.

firing_graph(S, eff)

Creates a digraph with divisors as vertices and edges between two divisors $$D$$ and $$E$$ if firing a single vertex in $$D$$ gives $$E$$.

INPUT:

S – Sandpile eff – list of divisors

OUTPUT:

DiGraph

EXAMPLES:

sage: S = sandpiles.Cycle(6)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div()
sage: firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01)


parallel_firing_graph(S,eff)

Creates a digraph with divisors as vertices and edges between two divisors D and E if firing all unstable vertices in D gives E.

INPUT:

S - Sandpile eff - list of divisors

OUTPUT:

DiGraph

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(6),0)
sage: D = SandpileDivisor(S, [1,1,1,1,2,0])
sage: eff = D.effective_div()
sage: parallel_firing_graph(S,eff).show3d(edge_size=.005,vertex_size=0.01)


random_DAG(num_verts,p=1/2,weight_max=1)

Returns a random directed acyclic graph with num_verts vertices. The method starts with the sink vertex and adds vertices one at a time. Each vertex is connected only to only previously defined vertices, and the probability of each possible connection is given by the argument p. The weight of an edge is a random integer between 1 and weight_max.

INPUT:

• num_verts - positive integer
• p - number between $$0$$ and $$1$$
• weight_max – integer greater than $$0$$

OUTPUT:

directed acyclic graph with sink $$0$$

EXAMPLES:

sage: S = random_DAG(5, 0.3)


sandpiles

Some examples of sandpiles.

Here are the available examples; you can also type “sandpiles.” and hit tab to get a list:

• “Complete()”
• “Cycle()”
• “Diamond()”
• “Grid()”
• “House()”

EXAMPLES:

sage: s = sandpiles.Complete(4)
sage: s.invariant_factors()
[1, 4, 4]
sage: s.laplacian()
[ 3 -1 -1 -1]
[-1  3 -1 -1]
[-1 -1  3 -1]
[-1 -1 -1  3]


wilmes_algorithm(M)

Computes an integer matrix L with the same integer row span as M and such that L is the reduced laplacian of a directed multigraph.

INPUT:

M - square integer matrix of full rank

OUTPUT:

L - integer matrix

EXAMPLES:

sage: P = matrix([[2,3,-7,-3],[5,2,-5,5],[8,2,5,4],[-5,-9,6,6]])
sage: wilmes_algorithm(P)
[ 1642   -13 -1627    -1]
[   -1  1980 -1582  -397]
[    0    -1  1650 -1649]
[    0     0 -1658  1658]


NOTES:

The algorithm is due to John Wilmes.

Help¶

Documentation for each method is available through the Sage online help system:

sage: SandpileConfig.fire_vertex?
Base Class:     <type 'instancemethod'>
String Form:    <unbound method SandpileConfig.fire_vertex>
Namespace:      Interactive
File:           /usr/local/sage-4.7/local/lib/python2.6/site-packages/sage/sandpiles/sandpile.py
Definition:     SandpileConfig.fire_vertex(self, v)
Docstring:
Fire the vertex v.

INPUT:

v - vertex

OUTPUT:

SandpileConfig

EXAMPLES:

sage: S = Sandpile(graphs.CycleGraph(3), 0)
sage: c = SandpileConfig(S, [1,2])
sage: c.fire_vertex(2)
{1: 2, 2: 0}


Note

An alternative to SandpileConfig.fire_vertex? in the preceding code example would be c.fire_vertex?, if c is any SandpileConfig.

Enter Sandpile.help(), SandpileConfig.help(), and SandpileDivisor.help() for lists of available Sandpile-specific methods.

General Sage documentation can be found at http://doc.sagemath.org/html/en/.