Adrien Boulanger, Pierre Mathieu
Let $Γ$ be a countable group acting on a geodesic hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ which generates a non elementary semi-group. Under the necessary assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance of a random walk with driving measure $μ$.
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We discuss the escape rate of the Brownian motion on a hyperbolic space. We point out that the escape rate is determined by using the Brownian expression of the radial part and a generalized Kolmogorov's test for the one dimensional Brownian motion.
Estimates on Escape Times for the Elephant Random Walk
本文研究了Elephant随机游走的逃逸时间，推导了其尾部的紧指数界，并证明在扩散 regime 中其行为与传统对称随机游走一致。
Consider a simple symmetric random walk on the integer lattice $\mathbb{Z}$. Let $E(n)$ denote a favorite edge of the random walk at time $n$. In this paper, we study the escape rate of $E(n)$, and show that almost surely $\liminf_{n\to\infty}\frac{|E(n)|}{\sqrt{n}\cdot(\log n)^{-γ}}$ equals 0 if $γ\le 1$, and is infinity otherwise. We also obtain a law of the iterated logarithm for $E(n)$.
Rate of escape of the conditioned two-dimensional simple random walk
本文利用KMT型耦合，从连续类似物的渐近结果推导出避免固定有限集的二维简单随机游走逃逸速率的精确渐近估计。
We study biased random walk on subcritical and supercritical Galton-Watson trees conditioned to survive in the transient, sub-ballistic regime. By considering offspring laws with infinite variance, we extend previously known results for the walk on the supercritical tree and observe new trapping phenomena for the walk on the subcritical tree which, in this case, always yield sub-ballisticity. This is contrary to the walk on the supercritical tree which always has some ballistic phase.
We study nearest-neighbors branching random walks started from a point at the interior of a hypercube. We show that the probability that the process escapes the hypercube is monotonically decreasing with respect to the distance of its starting point from the boundary. We derive as a consequence that at all times the number of particles at a site is monotonically decreasing with respect to its distance from the starting point.
We use subgroup distortion to determine the rate of escape of a simple random walk on a class of polycyclic groups, and we show that the rate of escape is invariant under changes of generating set for these groups. For metabelian groups, we define a stronger form of subgroup distortion, which applies to non-finitely generated subgroups. Under this hypothesis, we compute the rate of escape for certain random walks on metabelian groups via a comparison to the toppling of a dissipative abelian sandpile.
We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is,formally speaking, the Doob's $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.
Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups
我们考虑有限字母表上所有单词的集合上的随机游动，使得在每一步中，只有当前单词的最后两个字母可以被修改，并且只有一个字母可以被邻接或删除。我们假设转移概率仅取决于当前单词的最后两个字母。此外，我们还考虑了特殊情况下的随机游动的自由积合并的有限群，产生在一个自然的方式从随机游动的单因素。本文的目的是使用不同的技术计算这些随机游动关于自然长度函数的逃逸率的几个等价公式。
The rotor walk on a graph is a deterministic analogue of random walk. Each vertex is equipped with a rotor, which routes the walker to the neighbouring vertices in a fixed cyclic order on successive visits. We consider rotor walk on an infinite rooted tree, restarted from the root after each escape to infinity. We prove that the limiting proportion of escapes to infinity equals the escape probability for random walk, provided only finitely many rotors send the walker initially towards the root. For i.i.d. random initial rotor directions on a regular tree, the limiting proportion of escapes is …
Rotor walk is a deterministic analogue of simple random walk. For any given graph, we construct a rotor configuration for which the escape rate of the corresponding rotor walk is equal to the escape rate of simple random walk, and thus answer a question of Florescu, Ganguly, Levine, and Peres (2014).
Rate of Escape on Free Products
假设给定有限族或可数集$(V_i)_{i\in\mathcal{I}}$的自由积$V$和每个$V_i$上控制其上随机游动的概率测度，我们考虑自由积上的瞬态随机游动自然产生于$V_i$上的随机游动.本文证明了关于分组长度的逃逸率的存在性，即随机游动逃逸到无穷远的速度，并给出了逃逸率的计算公式.为此，我们提出了三种不同的方法，给出了三个不同的等价公式.
Rotor walk is a deterministic analogue of random walk. We study its recurrence and transience properties on Z^d for the initial configuration of all rotors aligned. If n particles in turn perform rotor walks starting from the origin, we show that the number that escape (i.e., never return to the origin) is of order n in dimensions d>=3, and of order n/log(n) in dimension 2.
Let $\{A, B, C\}$ be a partition of a sample space $Ω$. For a random walk $S_n = x + \sum_{j=1}^n X_j$ starting at $x \in A$, we find estimates for the Green's function $G_{A \cup B}(x,y)$ and the hitting time $E^x(T_C)$ for $x, y \in A \cup B$, with interest in the case where $C$ "separates" $A$ and $B$ in a sense (e.g. the probability of jumping from $A$ to $B$, or vice versa, before hitting $C$, is small).
We identify a fundamental phenomenon of heterogeneous one dimensional random walks: the escape (traversal) time is maximized when the heterogeneity in transition probabilities forms a pyramid-like potential barrier. This barrier corresponds to a distinct arrangement of transition probabilities, sometimes referred to as the pendulum arrangement. We reduce this problem to a sum over products, combinatorial optimization problem, proving that this unique structure always maximizes the escape time. This general property may influence studies in epidemiology, biology, and computer science to better …
We consider random walks on a non-elementary hyperbolic group endowed with a word distance. To a probability measure on the group are associated two numerical quantities, the rate of escape and the entropy. On the set of admissible probability measures whose support is contained in a given finite set, we show that both quantities depend in an analytic way on the probability measure. Our spectral techniques also give a new proof of the central limit theorem, and imply that the corresponding variance is analytic.
We prove that the trace of a transient branching random walk on a planar hyperbolic Cayley graph has a.s. continuum many ends and no isolated end.
Recurrence, Transience, and the Rate of Escape for Elephant Random Walks with Two Memory Channels
研究双记忆通道象随机游走，证明p=11/16为精确递归-暂态阈值，并确定暂态区间的逃逸速率及p=7/8时的逃逸尺度。
Acceleration of Lamplighter Random Walks
假设给定一个无限的非有限生成群$G$和一个在其圈积$(\mathbb{Z}/ 2\mathbb{Z})\wr G$上的瞬态随机游动，使得它在$G$上的投影是瞬态的，并且具有有限的一阶矩。这种随机游动可以解释为$G$上的点灯随机游动。我们的目的是表明，随机游动的圈积逃逸到无穷大的一个合适的（伪）度量快于其投影到$G$。我们还解决的情况下，伪度量是最短的“旅行推销员之旅”的长度。在此上下文中，排除某些退化情况，如果$G=\mathbb{Z}$，线性逃逸率严格大于点灯随机游动在$G$上投影的逃逸率。
Expected ballisticity of a continuous self avoiding walk on hyperbolic spaces $\mathbb{H}^d$ is established.
We study the simple random walk on stochastic hyperbolic half planar triangulations constructed in Angel and Ray [3]. We show that almost surely the walker escapes the boundary of the map in positive speed and that the return probability to the starting point after n steps scales like $\exp(-cn^{1/3})$.
We consider nonelementary random walks on general hyperbolic spaces. Without any moment condition on the walk, we show that it escapes linearly to infinity, with exponential error bounds. We even get such exponential bounds up to the rate of escape of the walk. Our proof relies on an inductive decomposition of the walk, recording times at which it could go to infinity in several independent directions, and using these times to control further backtracking.
Mean first escape times of Brownian motion on asymptotically hyperbolic and gas giant metric surfaces
研究了布朗运动在渐近双曲和气态巨行星度量表面的平均首次逃逸时间，通过渐近展开和数值模拟验证了不同度量下逃逸时间的行为差异。
We study random walks on relatively hyperbolic groups whose law is convergent, in the sense that the derivative of its Green function is finite at the spectral radius.When parabolic subgroups are virtually abelian, we prove that for such a random walk satisfies a local limit theorem of the form $p_n(e, e)\sim CR^{-n}n^{-d/2}$, where $p_n(e, e)$ is the probability of returning to the origin at time $n$, $R$ is the inverse of the spectral radius of the random walk and $d$ is the minimal rank of a parabolic subgroup along which the random walk is spectrally degenerate.This concludes the classific…
Random walks on dynamic graphs have received increasingly more attention from different academic communities over the last decade. Despite the relatively large literature, little is known about random walks that construct the graph where they walk while moving around. In this paper we study one of the simplest conceivable discrete time models of this kind, which works as follows: before every walker step, with probability $p$ a new leaf is added to the vertex currently occupied by the walker. The model grows trees and we call it the Bernoulli Growth Random Walk (BGRW). We show that the BGRW wa…
Continuous-time weakly self-avoiding walk on $\mathbb{Z}$ has strictly monotone escape speed
研究连续时间弱自避行走在整数格上的逃逸速度，通过传递矩阵、超对称BFS-Dynkin同构、谱理论、Tauber理论和随机占优方法，证明了逃逸速度严格随排斥强度增加。
Limit Theorems for Random Walks in the Hyperbolic Space
本文利用球模型及 Möbius 加法与乘法，证明了维数 n ≥ 2 的 Poincaré 双曲空间上随机游走的中心极限定理、局部极限定理，并导出相应大数定律。
Martin boundaries of diffusions and random walks on hyperbolic spaces
本文研究双曲空间上特定扩散过程及其在等距群上关联随机游走的Martin边界。
Random walk on hyperbolic groups and proper powers
本文研究了双曲群上的对称随机游走，证明到达真幂的概率与返回单位元的概率具有相同的指数衰减率。
In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate of escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to the generalised word length is derived. Finally, we show that the rate of escape, which can be regarded as a function in the finitely many parameters which describe the random walk, behaves as a real-analytic function in terms of probability measures of constant support.
We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In a second part of the paper, we show that the (exponential) deviation inequality holds for me…
Scaling limit of critical percolation clusters on hyperbolic random half-planar triangulations and the associated random walks
本文证明了半平面随机双曲三角剖分上临界渗流簇的Gromov-Hausdorff-Prohorov标度极限为布朗连续随机树，并推得其上的简单随机游走重标度为该树上的布朗运动。
We show that the asymptotic entropy of a random walk on a nonelementary hyperbolic group, with symmetric and bounded increments, is differentiable and we identify its derivative as a correlation. We also prove similar results for the rate of escape.
We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when the bias converges to its critical value is conjectured to coincide with the weak limit of the uniform SAW. Along the way, we obtain a criterion for the continuity of the escape probability of a biased random walk on a tree as a function of the bias, and show that the collection of escape probability functions for spherically symmetric trees of bounded degree…
We consider the operator associated to a random walk on finite volume surfaces with hyperbolic cusps. We study the spectral gap (upper and lower bound) associated to this operator and deduce some rate of convergence of the iterated kernel towards its stationary distribution.
Suppose we are given the free product V of a finite family of finite or countable sets. We consider a transient random walk on the free product arising naturally from a convex combination of random walks on the free factors. We prove the existence of the asymptotic entropy and present three different, equivalent formulas, which are derived by three different techniques. In particular, we will show that the entropy is the rate of escape with respect to the Greenian metric. Moreover, we link asymptotic entropy with the rate of escape and volume growth resulting in two inequalities.
Besov spaces and random walks on a hyperbolic group: boundary traces and reflecting extensions of Dirichlet forms
本文利用 Dirichlet 型理论证明低共形维数双曲群随机游走的无穷远迹过程存在并关联反射随机游走，同时建立相关 Besov 空间同构并分析调和测度正则性。
We show that the circle packing type of a unimodular random plane triangulation is parabolic if and only if the expected degree of the root is six, if and only if the triangulation is amenable in the sense of Aldous and Lyons. As a part of this, we obtain an alternative proof of the Benjamini-Schramm Recurrence Theorem. Secondly, in the hyperbolic case, we prove that the random walk almost surely converges to a point in the unit circle, that the law of this limiting point has full support and no atoms, and that the unit circle is a realisation of the Poisson boundary. Finally, we show that the…
Rotor walk is deterministic counterpart of random walk on graphs. We study that under a certain initial configuration in Z^d, n particles perform rotor walks from the origin consecutively. They would stop if they hit the origin or infinity. When the dimension d>=3, the escape rate exists and it attains the upper bound of O. Schramm. When the dimension d=2, the numbers of the particles escaping to infinity are of order n/log(n). The limit of their quotient exist and also attains the upper bound of L.Florescu,S.Ganguly,L.Levine,Y.Peres which equals to frac{pi}{2}. We use the results and the meth…
Rate of Escape on the Lamplighter Tree
假设我们有一个度为$q\geq 3$的齐次树$\mathcal{T}_q$，其中每个顶点都有一盏灯，可以打开或关闭。这个结构可以用圈积$(\mathbb{Z}/2)\wr Γ$来描述，其中$Γ=\ast_{i=1}^q \mathbb{Z}/2$是$q$因子$\mathbb{Z}/2$的自由积群。我们考虑$(\mathbb{Z}/2)\wr Γ$的Cayley图上的瞬态随机游动，我们要计算逃逸率的上下界上界，即随机游动逃逸到无穷大的速度。
Let $Γ$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $μ$ a probability measure on $Γ$ whose support generates a non-elementary subsemigroup. Under the assumption that $μ$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $μ$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.
We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (resp., rate of escape) of the transformed random walks is equal to the asymptotic entropy (resp., rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov's formula from ergodic theory, its particular cases were established earlier by Kaimanovich [1983] and Hartman, Lima, Tamuz [2014].
We consider nondegenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.
We show existence of the weak large deviation principle, with a convex rate function, for the renormalized distance from the starting point of irreducible random walks on relatively hyperbolic groups. Under the assumption of finiteness of exponential moments, the full large deviation principle holds, and the rate function governing it can be expressed as the Fenchel-Legendre transform of the limiting logarithmic moment generating function of the sequence of renormalized distances.
We show that simple random walks on (non-trivial) relatively hyperbolic groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy paths. Along the way, we also prove a refinement of the result that mapping class groups have quadratic divergence. An application of our theorem for relatively hyperbolic groups is that random triangles in non-trivial relatively hyperbolic groups are $O(\log(n))$-thin, random points have $O(\lo…
Central limit theorem and geodesic tracking on hyperbolic spaces and Teichmüller spaces
本文研究双曲空间和Teichmüller空间等距群上的随机游走，证明了平移长度满足中心极限定理的充要条件为二阶矩有限，并建立了测地线跟踪的次线性与对数界。
We are interested in predator-prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase-escape process, introduced by Kordzakhia with a variant by Bordenave, and the birth-and-assassination process, introduced by Aldous & Krebs. We exhibit a coupling between these processes and branching random walks starting from a random point and killed at the barrier 0. This sheds new light on the chase-escape and birth-and-a…
Given a probability measure on a finitely generated group, its Martin boundary is a natural way to compactify the group using the Green function of the corresponding random walk. For finitely supported measures in hyperbolic groups, it is known since the work of Ancona and Gou{ë}zel-Lalley that the Martin boundary coincides with the geometric boundary. The goal of this paper is to weaken the finite support assumption. We first show that, in any non-amenable group, there exist probability measures with exponential tails giving rise to pathological Martin boundaries. Then, for probability measur…
Continuity for random walks in hyperbolic percolation: drift, entropy, and dimension
本文证明了双曲群Cayley图渗流无限簇上随机游走的漂移和渐近熵随渗流参数连续变化，进而推出调和测度Hausdorff维数的连续性。
Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces
该研究针对Gromov双曲空间等距群上的随机游走，证明无矩条件下非初等随机游走平移长度至少线性增长，有限一阶矩下增长率等于漂移，还将技术推广到Teichmüller空间，在有限一阶矩下证明谱定理，拓展了已有结果。
Isotropic random walks on affine buildings
本文应用球调和分析的技巧证明了不可约型正则建筑物上各向同性随机游动的局部极限定理、逃逸率定理和中心极限定理。
Ratio-limit boundaries for random walks on relatively hyperbolic groups
本文研究相对双曲群上随机游走的比率极限边界，扩展了Woess策略并揭示其局限，证明谱非退化情形下极小点闭包为唯一最小不变子空间，并推广了Toeplitz C*-代数共泛商的存在性，且构造了反例。
We show that the displacement and translation distance of non-elementary random walks on isometry groups of hyperbolic spaces satisfy large deviation principles with the same rate function $I$. Roughly, this means that there exists function $I(t)$ which accurately predicts the exponential decay rate of the probability that the translation distance of a random product of length $n$ is $tn$, and similarly for the displacement. This settles a special case of a conjecture concerning the large deviation principle for the spectral radius of random matrix products. In a second part, we give a charact…
A Branching Process with Geometric Emigration and the "Have Your Cookie and Eat It" Random Walk
本文通过引入具有几何迁出的分支过程，研究了一类偏向随机游走的速度与逃逸概率，证实了Pinsky猜想，并计算了相关自交互随机游走的速度与逃逸概率。
We prove that for the $d$-regular tessellations of the hyperbolic plane by $k$-gons, there are exponentially more self-avoiding walks of length $n$ than there are self-avoiding polygons of length $n$. We then prove that this property implies that the self-avoiding walk is ballistic, even on an arbitrary vertex-transitive graph. Moreover, for every fixed $k$, we show that the connective constant for self-avoiding walks satisfies the asymptotic expansion $d-1-O(1/d)$ as $d\to \infty$; on the other hand, the connective constant for self-avoiding polygons remains bounded. Finally, we show for all …
We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in arXiv:1710.03137, and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the un…
Let $Γ$ be a non-elementary relatively hyperbolic group with a finite generating set. Consider a finitely supported admissible and symmetric probability measure $μ$ on $Γ$ and a probability measure $ν$ on $\mathbb{N}$ with mean $r$. Let $\mathrm{BRW}(Γ,ν,μ)$ be the branching random walk on $Γ$ with offspring distribution $ν$ and base motion given by the random walk with step distribution $μ$. It is known that for $1 < r \leq R$ with $R$ the radius of convergence for the Green function of the random walk, the population of $\mathrm{BRW}(Γ,ν,μ)$ survives forever, but eventually vacates every fin…
In this article we define and study a stochastic process on Galoisian covers of compact manifolds. The successive positions of the process are defined recursively by picking a point uniformly in the Dirichlet domain of the previous one. We prove a theorem à la Kesten for such a process: the escape rate of the random walk is positive if and only if the cover is non amenable. We also investigate more in details the case where the deck group is Gromov hyperbolic, showing the almost sure convergence to the boundary of the trajectory as well as a central limit theorem for the escape rate
Reduced Random Walks in the Hyperbolic Plane$\hspace{1pt}!\hspace{-3.8pt}?$
研究双曲三角群中的Lam约化随机游走，证明其几乎必然收敛，并为$PGL_2(\mathbb{Z})$情形下的极限分布给出了基于新定义的疑问感叹号函数的显式公式。
The ratio set of the harmonic measure of a random walk on a hyperbolic group
本文研究非顺从双曲群上有限程随机游走对应的Gromov边界调和测度的轨道等价关系，通过Martin核生成值确定其比值集，证明该等价关系为III型但绝非III₀型。
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