# Questions tagged [sandpile]

The sandpile tag has no usage guidance.

17
questions

**6**

votes

**2**answers

415 views

### 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 $f$ ...

**2**

votes

**1**answer

143 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 ...

**4**

votes

**0**answers

114 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 $Pic^0(G)$...

**13**

votes

**0**answers

464 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

**1**answer

466 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

**0**answers

101 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

**1**answer

330 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**

votes

**1**answer

401 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

**1**answer

260 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**

votes

**3**answers

629 views

### Sandpile group corresponding to Abelian group

How we can prove each finite Abelian group is the sandpile group for some graph ?

**6**

votes

**0**answers

179 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**

votes

**0**answers

310 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 ...

**41**

votes

**1**answer

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**

votes

**0**answers

504 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 ...

**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 ...

**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 ...

**6**

votes

**0**answers

250 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 ...