A proof of the Lyons-Pemantle-Peres monotonicity conjecture for high biases

An Elementary Proof of Hawkes's Conjecture on Galton-Watson Trees

In 1981, J. Hawkes conjectured the exact form of the Hausdorff gauge function for the boundary of supercritical Galton-Watson trees under a certain assumption on the tail at the infinity of the total mass of the branching measure. Hawkes's conjecture has been proved by T. Watanabe in 2007 as well as other other precise results on fractal properties of the boundary of Galton-Watson trees. The goal of this paper is to provide an elementary proof of Hawkes's conjecture under a less restrictive assumption than in T. Watanabe's paper, by use of size-biased Galton-Watson trees introduced by Lyons, P…

Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant.

A central limit theorem for random walk in random environment on marked Galton-Watson trees

In this article we focus on a general model of random walk on random marked trees. We prove a recurrence criterion, analogue to the recurrence criterion proved by R. Lyons and Robin Pemantle (1992) in a slightly different model. In the critical case, we obtain a criterion for the positive/null recurrence. Several regimes appear, as proved (in a similar model), by Y. Hu and Z. Shi (2007). We focus on the "diffusive" regime and improve their result in this case, by obtaining a functional Central Limit Theorem. Our result is also an extension of a result by Y. Peres and O. Zeitouni (2008), obtain…

Discretization methods for homogeneous fragmentations

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogs of a certain type of branching random walks, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations. On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, nam…

A subdiffusive behaviour of recurrent random walk in random environment on a regular tree

We are interested in the random walk in random environment on an infinite tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion. Our paper focuses on the almost sure asymptotic behaviours of a recurrent random walk $(X\_n)$ in random environment on a regular tree, which is closely related to Mandelbrot [13]'s multiplicative cascade. We prove, under some general assumptions upon the distribution of the environment, the existence of a new exponent $ν\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$ behaves asymptotically like $n^ν$. The value of $ν$ is explicitl…

General random walk in a random environment defined on Galton-Watson trees

We consider the motion of a particle on a Galton Watson tree, when the probabilities of jumping from a vertex to any one of its neighbours is determined by a random process. Given the tree, positive weights are assigned to the edges in such a way that, viewed along any line of descent, they evolve as a random process. In order to introduce our method for proving transience or recurrence, we first suppose that the weights are i.i.d., reproving a result of Lyons and Pemantle. We then extend the argument to allow a Markovian environment, and finally to a random walk on a Markovian environment tha…

Harmonic measure for biased random walk in a supercritical Galton-Watson tree

We consider random walks $λ$-biased towards the root on a Galton-Watson tree, whose offspring distribution $(p_k)_{k\geq 1}$ is non-degenerate and has finite mean $m>1$. In the transient regime $0<λ<m$, the loop-erased trajectory of the biased random walk defines the $λ$-harmonic ray, whose law is the $λ$-harmonic measure on the boundary of the Galton-Watson tree. We answer a question of Lyons, Pemantle and Peres by showing that the $λ$-harmonic measure has a.s. strictly larger Hausdorff dimension than the visibility measure, which is the harmonic measure corresponding to the simple forward ra…

Painting a graph with competing random walks

Let $X_1,X_2$ be independent random walks on $\mathbf{Z}_n^d$, $d\geq3$, each starting from the uniform distribution. Initially, each site of $\mathbf{Z}_n^d$ is unmarked, and, whenever $X_i$ visits such a site, it is set irreversibly to $i$. The mean of $|\mathcal{A}_i|$, the cardinality of the set $\mathcal{A}_i$ of sites painted by $i$, once all of $\mathbf{Z}_n^d$ has been visited, is $\frac{1}{2}n^d$ by symmetry. We prove the following conjecture due to Pemantle and Peres: for each $d\geq3$ there exists a constant $α_d$ such that $\lim_{n\to\infty}\operatorname{Var}(|\mathcal {A}_i|)/h_d(…

簇图的应用

Applications of the cluster graphing

该研究利用可数Borel等价关系理论与Gaboriau的簇图构造，简短证明了1999年关于单模非可和准传递图上伯努利渗流临界值处无无限簇的结果，并提及簇图构造的其他应用，旨在普及专家熟知的民间证明。

Cutpoints and resistance of random walk paths

We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also prove that the expected number of cutpoints of any transient Markov chain is infinite. This answers two questions of James, Lyons and Peres [A Transient Markov Chain With Finitely Many Cutpoints (2007) Festschrift for David Freedman]. Additionally, we consider a simple random walk on a finite connected graph $G$ that starts at some fixed vertex $x$ and is sto…

随机最小生成树中的成对边相关性：通用界与完全图负相关性

Pairwise edge correlations in random minimum spanning trees: a universal bound and complete-graph negative correlation

本文针对随机最小生成树的成对边相关性，证明正相关性受通用界控制、$n\boldsymbol{\u2265}3$时完全图边对负相关，还给出边分布非相同时无通用常数的反例。

A symmetric entropy bound on the non-reconstruction regime of Markov chains on Galton-Watson trees

We give a criterion of the form Q(d)c(M)<1 for the non-reconstructability of tree-indexed q-state Markov chains obtained by broadcasting a signal from the root with a given transition matrix M. Here c(M) is an explicit function, which is convex over the set of M's with a given invariant distribution, that is defined in terms of a (q-1)-dimensional variational problem over symmetric entropies. Further Q(d) is the expected number of offspring on the Galton-Watson tree. This result is equivalent to proving the extremality of the free boundary condition-Gibbs measure within the corresponding Gibbs…

群上随机游走的小球估计

Small-ball estimates for random walks on groups

本文证明了一个新不等式，将群上随机游走的小位移概率与谱和等周轮廓关联，并由此在Lyons等人关于暂态随机游走占据时间的猜想上取得进展，同时给出强定量小球估计，并验证了若干经典例子的正则性条件。