Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

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 sing…

Recovering a tree from the lengths of subtrees spanned by a randomly chosen sequence of leaves

Given an edge-weighted tree $T$ with $n$ leaves, sample the leaves uniformly at random without replacement and let $W_k$, $2 \le k \le n$, be the length of the subtree spanned by the first $k$ leaves. We consider the question, "Can $T$ be identified (up to isomorphism) by the joint probability distribution of the random vector $(W_2, \ldots, W_n)$?" We show that if $T$ is known {\em a priori} to belong to one of various families of edge-weighted trees, then the answer is, "Yes." These families include the edge-weighted trees with edge-weights in general position, the ultrametric edge-weighted …

Maximal local time of randomly biased random walks on a Galton-Watson tree

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.

A time-invariant random graph with splitting events

We introduce a process where a connected rooted multigraph evolves by splitting events on its vertices, occurring randomly in continuous time. When a vertex splits, its incoming edges are randomly assigned between its offspring and a Poisson random number of edges are added between them. The process is parametrised by a positive real $λ$ which governs the limiting average degree. We show that for each value of $λ$ there is a unique random connected rooted multigraph $M(λ)$ invariant under this evolution. As a consequence, starting from any finite graph $G$ the process will almost surely conver…

微醉警察与微醉强盗：图上有偏随机游走的碰撞

Tipsy cop and tipsy robber: collisions of biased random walks on graphs

本文重新定义微醉警察与微醉强盗博弈的四种回合结果，并在无限网格和多类无限树上分析双方策略。

Recurrence of Distributional Limits of Finite Planar Graphs

Suppose that $G_j$ is a sequence of finite connected planar graphs, and in each $G_j$ a special vertex, called the root, is chosen randomly-uniformly. We introduce the notion of a distributional limit $G$ of such graphs. Assume that the vertex degrees of the vertices in $G_j$ are bounded, and the bound does not depend on $j$. Then after passing to a subsequence, the limit exists, and is a random rooted graph $G$. We prove that with probability one $G$ is recurrent. The proof involves the Circle Packing Theorem. The motivation for this work comes from the theory of random spherical triangulatio…