Questions tagged [sandpile]

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Measuring the failure of basepoint independence of the rotor-routing model for non-planar ribbon graphs

In this question from 2012, Jordan Ellenberg asks if the set of spanning trees of a graph $G$ is naturally a torsor for the critical group (also called the sandpile group or the picard group $Pic^0(G)$...
12
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0answers
422 views

Strange formula in arithmetic dynamic

Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two. We discovered the following operator which acts on the space of polynomials (or ...
4
votes
1answer
348 views

How is the Jacobian or Sandpile group of a graph computed?

From what I understand, given a graph, the Jacobian group and the Sandpile group refer to the same object. Until now, I have been computing this group in the way detailed in Chapter 1 of this ...
5
votes
0answers
97 views

Probabilistic distribution of sandpile model type

Let $G=(V,E)$ be a connected graph. Assume that $m\leqslant |V|$ hedgehogs sit in the vertices of $G$. If there are $r\geqslant 2$ hedgehogs in the same vertex $v\in V$, one of them goes to a randomly ...
5
votes
1answer
288 views

power laws emerging from the sandpile model

Is there a rigorous proof that the abelian sandpile model generates a power law distribution of avalanche lengths?
7
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1answer
387 views

Flooding a cycle digraph via chip-firing: $n^{k-1} + n^{k-2} + \cdots + 1$ bound (a Norway 1998-99 problem generalized)

Let $k > 1$ and $n$ be positive integers. Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Let $D$ be a digraph which has exactly $k$ vertices $v_0$, $v_1$, ..., $v_{k-1}$ and exactly $k$ arcs $v_0 \...
5
votes
1answer
249 views

Duration and critical groups order in sandpile models and chip firing games

The famous chip firing game (which is closely related to sandpile models) goes like this: Place chips at the vertices of a graph. REPEATEDLY: If a vertex $v$ of degree $d_{v}$ has at least $d_{v}...
4
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3answers
597 views

Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?
6
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0answers
177 views

Local structure in the stochastic sandpile model

Here's a question that came up at the recent AIM conference on chip-firing and generalizations. The stochastic sandpile model, I think originally due to Manna, is a stochastic process that (in one ...
4
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0answers
300 views

What is the graphical version of the circle parking story?

The classical parking function story is as follows: we have cars $v_1,\ldots,v_n$ who approach a line of spaces marked $0,\ldots,n-1$ in order. Each car $v_i$ has space preference $a_i$. A car will ...
40
votes
1answer
3k views

What is the sandpile torsor?

Let G be a finite undirected connected graph. A divisor on G is an element of the free abelian group Div(G) on the vertices of G (or an integer-valued function on the vertices.) Summing over all ...
10
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0answers
497 views

Abelian sandpile models

This question is about a popular probabilistic model on graphs studied in physics, mostly, for the standard lattice in ${\mathbb R}^n$ but also on other graphs (this model is of the same spirit as ...
11
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2answers
1k views

Why is the identity element of the sandpile group self-similar?

I've been reading about the Abelian Sandpile Model and noticed the identity element of the sandpile group on the square has self-similar components. The sandpile group of the 198x198 square of ...
26
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1answer
2k views

Who wins this two-player game based on the sandpile model?

Given a connected graph $G$, two players, Blue and Green, play the following game: initially, all vertices are unclaimed. Players alternate turns. On her turn, Blue adds a token to either an ...
6
votes
0answers
245 views

subrandom walkers

Does anyone know of any work on the following model or variants thereof?: Finitely many chips are distributed on the integers at time 0. To find the distribution at time $t+1$, take all the chips at ...