随机有根图上的阿贝尔沙堆模型和自相似群
Abelian Sandpile Model on Randomly Rooted Graphs and Self-Similar Groups
摘要：
本文的主要结果是一个严格的临界性证明和显式计算的临界指数的衰减的阿贝尔沙堆模型（ASM）上的一个大家庭的无限图。首先引入了有限图极限的ASM在局部收敛中的临界性概念，这就引出了ASM在随机弱极限中的临界性问题。我们的主要技术成分是有限仙人掌序列上ASM的几乎必然临界性的充分条件（即，可分图的块是圈或单边）的假设下，随机弱极限几乎必然是1-结束。允许显式计算临界指数的例子来自有限生成群在规则有根树上通过自同构的作用。限制行动的连续水平的树定义了一个序列的有限图，其限制在局部收敛的轨道Schreier图的行动组的边界上的树。在复多项式的迭代单值群的情况下，这些图是仙人掌，我们表明，对于1-结束的，临界指数的雪崩的质量衰减依赖于图形的增长。与$z^2-1$相关的著名的Basilica群产生了无数个4正则单端二次增长图，其雪崩质量的临界指数等于1;以及无数个新的非临界二次增长的例子（因此不是$\mathbb{Z}$的准等距）。另一个迭代单值群，我们认为提供了不可数许多图的临界指数等于$2\log 2/\log 3 >1$。最后，我们还展示了图的多项式增长任意小的临界指数。
英文摘要：
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.
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