Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups   The main result of this paper is a rigorous proof of criticality and an
explicit computation of critical exponents for the decay of avalanches in the
Abelian sandpile model (ASM) on a large family of infinite graphs. We begin by
introducing the notion of criticality of the ASM for limits of finite graphs in
local convergence, which naturally leads to the question about criticality of
the ASM in the random weak limit. Our main technical ingredient is a sufficient
condition for almost sure criticality of the ASM on sequences of finite cacti
(i.e., separable graphs whose blocks are cycles or single edges) under the
assumption that the random weak limit is almost surely 1-ended. Examples that
allow explicit computations of the critical exponents come from actions of
finitely generated groups on regular rooted trees, by automorphisms.
Restricting the action to the consecutive levels of the tree defines a sequence
of finite graphs whose limits in the local convergence are orbital Schreier
graphs for the action of the group on the boundary of the tree. In the case of
iterated monodromy groups of complex polynomials, these graphs are cacti, and
we show that for 1-ended ones, the critical exponent for the decay of the mass
of avalanches depends on the growth of the graph. The well-known Basilica group
related to $z^2-1$ gives rise to uncountably many 4-regular one-ended graphs of
quadratic growth with the critical exponent for the mass of avalanches equal to
1; as well as uncountably many new non-critical examples of quadratic growth
(thus not quasi-isometric to $\mathbb{Z}$). Another iterated monodromy group
that we consider provides uncountably many graphs with the critical exponent
equal to $2\log 2/\log 3 >1$. Finally, we also exhibit graphs of polynomial
growth with arbitrarily small critical exponent.

30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn