Emilio Corso
We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.
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In this note, we generalise a Bourgain's construction of finitely-supported symmetric measures whose Furstenberg measure has a smooth density from the case of $\mathrm{SL}_2(\mathbb{R})$ to that of a general simple Lie group. The proof is the same as Bourgain's, except that the use of Fourier series is replaced by harmonic analysis on a maximal compact subgroup.
Vertex-reinforced random walk is defined in Pemantle's (1988) thesis; it is a random walk that is biased to visit sites it has already visited a lot. We show that this reinforcement scheme, in contrast to the scheme of edge-reinforcement, causes random walk on a line to get trapped in a finite set.
We present a class of graphs where simple random walk is recurrent, yet two independent walkers meet only finitely many times almost surely. In particular, the comb lattice, obtained from Z^2 by removing all horizontal edges off the X-axis, has this property. We also conjecture that the same property holds for some other graphs, including the incipient infinite cluster for critical percolation in Z^2.
Singularity of harmonic measure for finitely supported random walks
本文通过构造测地流并利用边界傅里叶分析，证明了余紧富克斯群上有限支撑非退化随机游走的调和测度具有奇异性。
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 give a solution to the inverse problem (given a function, find a corresponding group) for large classes of speed, entropy, isoperimetric profile, return probability and $L_p$-compression functions of finitely generated groups of exponential volume growth. For smaller classes, we give solutions among solvable groups. As corollaries, we prove a recent conjecture of Amir on joint evaluation of speed and entropy exponents and we obtain a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups, originally due to Cornulier and Tessera. We also obtain a formula …
Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.
We show that for each $λ\in [\frac{1}{2}, 1]$, there exists a solvable group and a finitely supported measure such that the associated random walk has upper speed exponent $λ$.
The speed of random walks on semigroups
本文构造了不同速度指数的有限生成半群随机游走，揭示其速度可任意慢却趋于无穷的特性，对比群论情形，并给出除退化情况外距离的通用下界。
For every $3/4\le δ, β< 1$ satisfying $δ\leq β< \frac{1+δ}{2}$ we construct a finitely generated group $Γ$ and a (symmetric, finitely supported) random walk $X_n$ on $Γ$ so that its expected distance from its starting point satisfies $E|X_n|\asymp n^β$ and its entropy satisfies $H(X_n)\asymp n^δ$. In fact, the speed and entropy can be set precisely to equal any two nice enough prescribed functions $f,h$ up to a constant factor as long as the functions satisfy the relation $n^{\frac{3}{4}}\leq h(n)\leq f(n)\leq \sqrt{{nh(n)}/{\log (n+1)}}\leq n^γ$ for some $γ<1$.
The predicable dance of random walk: local limit theorems on finitely-generated abelian groups
本文提出一种通用傅里叶分析方法描述有限生成阿贝尔群上的随机游走，在无非周期性和不可约性假设下证明了局部中心极限定理，将渐近行为分解为描述扩散和舞蹈的两个函数乘积。
We consider random walk on the group of uni-upper triangular matrices with entries in $\mathbb{F}_2$ which forms an important example of a nilpotent group. Peres and Sly (2013) proved tight bounds on the mixing time of this walk up to constants. It is well known that the single column projection of this chain is the one dimensional East process. In this article, we complement the Peres-Sly result by proving a cutoff result for the mixing of finitely many columns in the upper triangular matrix walk at the same location as the East process of the same dimension. Moreover, we also show that the s…
We obtain a large deviations principle for the self-intersection local times for a symmetric random walk in dimension d>4. As an application, we obtain moderate deviations for random walk in random sceneries in some region of parameters.
We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a collection of a random number of dependent random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns o…
We consider the branching random walk drifting to $-\infty$ and we investigate large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.
Record numbers are basic statistics in random walks, whose deviation principles are not very clear so far. In this paper, the asymptotic probabilities of large and moderate deviations for numbers of weak records in right continuous or left continuous random walks are proved.
Random walks and contracting elements I: Deviation inequality and Limit laws
本文研究具有压缩等距的度量空间上的随机游走，通过调整Gouëzel的关键时间构造建立尖锐偏差不等式，并应用于映射类群和CAT(0)空间，得到指数界、中心极限定理、重对数律和测地线跟踪等极限定律。
We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator-stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.
Consider the random walk $G_n : = g_n \ldots g_1$, $n \geq 1$, where $(g_n)_{n\geq 1}$ is a sequence of independent and identically distributed random elements with law $μ$ on the general linear group ${\rm GL}(V)$ with $V=\mathbb R^d$. Under suitable conditions on $μ$, we establish Cramér type moderate deviation expansions and local limit theorems with moderate deviations for the coefficients $\langle f, G_n v \rangle$, where $v \in V$ and $f \in V^*$. Our approach is based on the Hölder regularity of the invariant measure of the Markov chain $G_n \!\cdot \! x = \mathbb R G_n v$ on the projec…
In this paper, we establish the invariance principle and the large deviation for the biased random walk $RW_λ$ with $λ\in [0,1)$ on $\mathbb{Z}^d, d\geq 1$.
We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown.
Given a symmetric random walk in $Z^2$ with finite second moments, let $R_n$ be the range of the random walk up to time $n$. We study moderate deviations for $R_n -E R_n$ and $E R_n -R_n$. We also derive the corresponding laws of the iterated logarithm.
On large deviations for the range of a two-dimensional random walk
本文研究了二维随机游走范围的大偏差问题，计算了在特定偏差尺度下访问顶点数超过预期的概率。
In this paper, we consider the linearly reinforced and the once-reinforced random walk models in the transient phase on trees. We show the large deviations for the upper tails for both models. We also show the exponential decay for the lower tail in the once-reinforced random walk model. On the other hand, the lower tail is in polynomial decay for the linearly reinforced random walk model.
We answer an open question of Grigorchuk and Zuk about amenability using random walks. Our results separate the class of amenable groups from the closure of subexponentially growing groups under the operations of group extension and direct limits; these classes are separated even within the realm of finitely presented groups.
We consider the small deviation probability for random walk with time-inhomogeneous random environment. Compared with the result in Mogul'ski\uı (1974) for the i.i.d. random walk, the rate is smaller (due to the random environment), which is specified in terms of the quenched and annealed variance.
We obtain large deviations estimates for the self-intersection local times for a symmetric random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length $n$, comes from sites visited less than some power of $\log(n)$. This is opposite to the situation in dimensions larger or equal to 5. Finally, we present two applications of our estimates: (i) to moderate deviations estimates for the range of a random walk, and (ii) to moderate deviations for random walk in random sceneries.
It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviation scale which we call hidden large deviation. We apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of severe congestion times for (buffered) queues. We conclude with simulated examples to verify our results.
We establish large deviation principles and phase transition results for both quenched and annealed settings of nearest-neighbor random walks with constant drift in random nonnegative potentials on $\mathbb Z^d$. We complement the analysis of \cite{Zer}, where a shape theorem on the Lyapunov functions and a large deviation principle in absence of the drift are achieved for the quenched setting.
We study large deviations for random walks on Lie groups defined by $σ_n^n = \exp(\frac1nX_1)\cdots\exp(\frac1nX_n)$, where $\{X_n\}_{n\geq1}$ is an i.i.d sequence of bounded random variables in the Lie algebra $\mathfrak{g}$. We follow a similar approach as in the proof of large deviations for geodesic random walks as given in [Ver19]. This approach makes it possible to simply rescale the increments of the random walk, without having to resort to dilations in order to reduce the influence of higher order commutators. Finally, we will apply this large deviation result to the Lie group of stoch…
Precise upper deviation estimates for the maximum of a branching random walk
本文研究了分支随机游动最大位移的精确上大偏差估计，描述了该大偏差事件条件下的极值过程，并引入了类似分支布朗运动装饰测度的点测度族。
Let $G$ be a Cayley graph of a nonamenable group with spectral radius $ρ< 1$. It is known that branching random walk on $G$ with offspring distribution $μ$ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $\bar μ$ satisfies $\bar μ\leq ρ^{-1}$. Benjamini and Müller (2010) conjectured that throughout the transient supercritical phase $1<\barμ \leq ρ^{-1}$, and in particular at the recurrence threshold $\bar μ= ρ^{-1}$, the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost sur…
We introduce the discrete affine group of a regular tree as a finitely generated subgroup of the affine group. We describe the Poisson boundary of random walks on it as a space of configurations. We compute isoperimetric profile and Hilbert compression exponent of the group. We also discuss metric relationship with some lamplighter groups and lamplighter graphs.
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.
Quenched large deviations for randomly weighted geodesic random walks
本文研究了加权测地随机游走的大偏差问题，证明了其在适当缩放下满足大偏差原理，并扩展了之前在实值情况下的结果。
For every 3/4 <= beta < 1 we construct a finitely generated group so that the expected distance of the simple random walk from its starting point is within a constant factor of n^beta. In fact, the speed can be set precisely to equal any nice prescribed function up to a constant factor.
We study the large deviations of one-dimensional excited random walks. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $…
We study random walks on groups with the feature that, roughly speaking, successive positions of the walk tend to be "aligned". We formalize and quantify this property by means of the notion of deviation inequalities. We show that deviation inequalities have several consequences including Central Limit Theorems, the local Lipschitz continuity of the rate of escape and entropy, as well as linear upper and lower bounds on the variance of the distance of the position of the walk from its initial point. In a second part of the paper, we show that the (exponential) deviation inequality holds for me…
Non-conformality of large deviations of moving average of the random walk in strongly mixing environment
本文证明了在强混合环境中，随机游动的淬火与退火大偏差无论无序程度如何，总在自然区域的某内部点处不一致，推翻了低无序时的一致性结论。
Phase transition for recurrence of stationary random walks on lamplighter groups
本文研究灯谜群上平稳随机游走的常返性，证明当基群为 virtually-$\mathbb{Z}$ 且灯群有限时存在常返-暂态相变，并确定 $G=\mathbb{Z}$ 时的精确临界参数。
We consider a bounded step size random walk in an ergodic random environment with some ellipticity, on an integer lattice of arbitrary dimension. We prove a level 3 large deviation principle, under almost every environment, with rate function related to a relative entropy.
We obtain sharp upper and lower bounds for the moderate deviations of the volume of the range of a random walk in dimension five and larger. Our results encompass two regimes: a Gaussian regime for small deviations, and a stretched exponential regime for larger deviations. In the latter regime, we show that conditioned on the moderate deviations event, the walk folds a small part of its range in a ball-like subset. Also, we provide new path properties, in dimension three as well. Besides the key role Newtonian capacity plays in this study, we introduce two original ideas, of general interest, …
Sample Path Large Deviations for Random Walks on Regular Trees
本文研究正则树上最近邻随机游走的样本路径大偏差问题，建立了到根节点距离的样本路径大偏差原理，并通过Fenchel-Legendre变换给出速率函数的隐式表达式。
A non-symmetric Kesten criterion and ratio limit theorem for random walks on amenable groups
本文针对可数群上的非对称随机游走，推广了Kesten关于谱半径与可顺从性的等价准则，并证明了可顺从群的比率极限定理。
We study downward deviations of the boundary of the range of a transient walk on the Euclidean lattice. We describe the optimal strategy adopted by the walk in order to shrink the boundary of its range. The technics we develop apply equally well to the range, and provide pathwise statements for the {\it Swiss cheese} picture of Bolthausen, van den Berg and den Hollander \cite{BBH}.
We consider a $\mathbb{R}^d$-valued branching random walk with a stationary and ergodic environment $ξ=(ξ_n)$ indexed by time $n\in\mathbb{N}$. Let $Z_n$ be the counting measure of particles of generation $n$. With the help of the uniform convergence of martingale and the multifractal analysis, we establish a large deviation result for the measures $Z_n$ as well as a moderate deviation principle.
We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em pocket-extension} of a group $G$---this leads to the study of certain finitely generated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$. Some sharp estimates are obtained while many challenging questions remain.
We consider a random walk in a random environment (RWRE) on the strip of finite width $\mathbb{Z} \times \{1,2,\ldots,d\}$. We prove both quenched and averaged large deviation principles for the position and the hitting times of the RWRE. Moreover, we prove a variational formula that relates the quenched and averaged rate functions, thus extending a result of Comets, Gantert, and Zeitouni for nearest-neighbor RWRE on $\mathbb{Z}$
We derive properties of the rate function in Varadhan's (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
A variety of behaviors of entropy functions of random walks on finitely generated groups is presented, showing that for any $\frac{1}{2}\leq α\leqβ\leq1$, there is a group $Γ$ with measure $μ$ equidistributed on a finite generating set such that \[\liminf\frac{\log H_{Γ,μ}(n)}{\log n}=α,\qquad \limsup \frac{\log H_{Γ,μ}(n)}{\log n}=β.\] The groups involved are finitely generated subgroups of the group of automorphisms of an extended rooted tree. The return probability and the drift of a simple random walk $Y_n$ on such groups are also evaluated, providing an example of group with return probab…
Let $X$ be a Lévy process with regularly varying Lévy measure $ν$. We obtain sample-path large deviations for scaled processes $\bar X_n(t) \triangleq X(nt)/n$ and obtain a similar result for random walks. Our results yield detailed asymptotic estimates in scenarios where multiple big jumps in the increment are required to make a rare event happen; we illustrate this through detailed conditional limit theorems. In addition, we investigate connections with the classical large deviations framework. In that setting, we show that a weak large deviation principle (with logarithmic speed) holds, but…
Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements of GL d (R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of order p \textgreater{} 1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.
Let B_n be the number of self-intersections of a symmetric random walk with finite second moments in the integer planar lattice. We obtain moderate deviation estimates for B_n - E B_n and E B_n- B_n, which are given in terms of the best constant of a certain Gagliardo-Nirenberg inequality. We also prove the corresponding laws of the iterated logarithm.
Our model is a constrained homogeneous random walk in a nonnegative orthant Z_+^d. The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed by Meyn and Tweedie. In this paper we show that, for this type of random walks, computing the stationary probability exactly is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this mo…
We consider non-degenerate, finitely supported random walks on a free group. We show that the entropy and the linear drift vary analytically with th eprobability of constant support.
Large deviation principle for empirical measures of once-reinforced random walks on finite graphs
本文研究有限图上一次强化随机游走的大偏差原理，通过改进的弱收敛方法证明了大偏差原理，并发现当δ=1时出现相变。
A strong large deviation principle for the empirical measure of random walks
本文证明了连续时间随机游走（特别是简单随机游走）的经验测度在 Mukherjee-Varadhan 拓扑下满足强大偏差原理，补充了布朗运动情形的大偏差结果。
Let $(X_t,t\geq 0)$ be a random walk on $\mathbb{Z}^d$. Let $ l_T(x)= \int_0^T δ_x(X_s)ds$ the local time at the state $x$ and $ I_T= \sum\limits_{x\in\mathbb{Z}^d} l_T(x)^q $ the q-fold self-intersection local time (SILT). In \cite{Castell} Castell proves a large deviations principle for the SILT of the simple random walk in the critical case $q(d-2)=d$. In the supercritical case $q(d-2)>d$, Chen and Mörters obtain in \cite{ChenMorters} a large deviations principle for the intersection of $q$ independent random walks, and Asselah obtains in \cite{Asselah5} a large deviations principle for the…
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.
Consider a random walk in a time-dependent random environment on the lattice Zd. Recently, Rassoul-Agha, Seppalainen and Yilmaz [RSY11] proved a general large deviation principle under mild ergodicity assumptions on the random environment for such a random walk, establishing first level 2 and 3 large deviation principles. Here we present two alternative short proofs of the level 1 large deviations under mild ergodicity assumptions on the environment: one for the continuous time case and another one for the discrete time case. Both proofs provide the existence, continuity and convexity of the…
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