Questions tagged [sandpile]
The sandpile tag has no usage guidance.
17
questions
7
votes
2
answers
450
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
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 ...
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 $Pic^0(G)$...
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 ...
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 ...
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 ...
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?
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 \...
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}...
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 ?
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 ...
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 ...
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 ...
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 ...
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 ...
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
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 ...