Questions tagged [sandpile]

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Chip-firing clocks

Let $G$ be some outdegree-regular directed graph with $n$ vertices and let $H$ be the Laplacian of $G$, so that the rows of $H$ correspond to chip-firing moves. I’m interested in linear functions $f$ ...
James Propp's user avatar
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2 votes
1 answer
174 views

Generators of sandpile groups of wheel graphs

In the paper "On the Sandpile Group of a Graph" by Cori and Rossin one can find a result related to the structure of the sandpile group of $W_n$. Is there a way to provide a set of ...
castor's user avatar
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4 votes
0 answers
124 views

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 $Pic^0(G)$...
Jonathan Gerhard's user avatar
13 votes
0 answers
491 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 ...
Nikita Kalinin's user avatar
5 votes
1 answer
638 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 ...
Aaron Bagheri's user avatar
6 votes
0 answers
107 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 ...
Fedor Petrov's user avatar
5 votes
1 answer
352 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?
Felix Goldberg's user avatar
7 votes
1 answer
417 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 \...
darij grinberg's user avatar
5 votes
1 answer
275 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}...
Felix Goldberg's user avatar
4 votes
3 answers
675 views

Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?
Hamed Khalilian's user avatar
6 votes
0 answers
182 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 ...
JSE's user avatar
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4 votes
0 answers
331 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 ...
Sam Hopkins's user avatar
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51 votes
3 answers
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 ...
JSE's user avatar
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10 votes
0 answers
522 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 ...
user avatar
12 votes
3 answers
2k 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 height ...
john mangual's user avatar
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25 votes
1 answer
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 ...
JBL's user avatar
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6 votes
0 answers
261 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 ...
James Propp's user avatar
  • 19.4k