We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in \cite[Section 1,2]{BJKS}, and in particular, imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long range percolation cluster.
We show that the trace of the null recurrent biased random walk on a Galton-Watson tree properly renormalized converges to the Brownian forest. Our result extends to the setting of the random walk in random environment on a Galton-Watson tree.
On a locally compact group $E$ with countable base, we consider a random walk $X$ that has a unique (up to a positive factor) $r$-invariant measure for some $r>0$. Under some weak conditions on the measure, there is a unique continuous exponential on $E$ naturally associated with $X$. It follows that there exists an $R$-recurrent random walk in the sense of Tweedie on $E$ if and only if $E$ is a recurrent group and there exists a Harris random walk on~$E$.
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 prove that vertex-reinforced random walk on the integers with weight of order k to the power alpha, for alpha in [0, 1/2), is recurrent. This confirms a conjecture of Volkov for alpha<1/2. The conjecture for alpha in [1/2, 1) remains open.
We consider d independent walkers on Z, m of them performing simple symmetric random walk and r = d -- m of them performing recurrent RWRE (Sinai walk), in I independent random environments. We show that the product is recurrent, almost surely, if and only if m $\le$ 1 or m = d = 2. In the transient case with r $\ge$ 1, we prove that the walkers meet infinitely often, almost surely, if and only if m = 2 and r $\ge$ I = 1. In particular, while I does not have an influence for the recurrence or transience, it does play a role for the probability to have infinitely many meetings. To obtain these …
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.
In this paper we are interested in a random walk in a random environment on a super-critical Galton-Watson tree. We focus on the recurrent cases already studied by Y. Hu and Z. Shi and G. Faraud. We prove that the largest generation entirely visited by these walks behaves like log n and that the constant of normalization which differs from a case to another is function of the inverse of the constant of Biggins' law of large number for branching random walks.
We clarify the asymptotic of the limsup of the size of the neighborhood of concentration of Sinai's walk improving the result in \cite{Pierre3}. Also we get the almost sure limit of the number of points visited more than a small but fixed proportion of a given amount of time.
For any recurrent random walk (S_n)_{n>0} on R, there are increasing sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_{n>0} has at least one finite accumulation point. For one class of random walks, we give a criterion on (g_n)_{n>0} and the distribution of S_1 determining the set of accumulation points for (g_n S_n)_{n>0}. This extends, with a simpler proof, a result of K.L. Chung and P. Erdos. Finally, for recurrent, symmetric random walks, we give a criterion characterizing the increasing sequences (g_n)_{n>0} of positive numbers for which liminf g_n|S_n|=0.
We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.
We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results of D. Dolgopyat and E. Kosygina for excited random walks with "boundedly many cookies per site." In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious w…
We consider recurrent diffusive random walks on a strip. We present constructive conditions on Green functions of finite sub-domains which imply a Central Limit Theorem with polynomial error bound, a Local Limit Theorem, and mixing of environment viewed by the particle process. Our conditions can be verified for a wide class of environments including independent environments, quasiperiodic environments, and environments which are asymptotically constant at infinity. The conditions presented deal with a fixed environment, in particular, no stationarity conditions are imposed.
Simple random walks on a partially directed version of $\mathbb{Z}^2$ are considered. More precisely, vertical edges between neighbouring vertices of $\mathbb{Z}^2$ can be traversed in both directions (they are undirected) while horizontal edges are one-way. The horizontal orientation is prescribed by a random perturbation of a periodic function, the perturbation probability decays according to a power law in the absolute value of the ordinate. We study the type of the simple random walk, i.e.\ its being recurrent or transient, and show that there exists a critical value of the decay power, ab…
In this paper we consider an irreducible random walk on the integer lattice $\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable process with index $α\in (1, 2)$ and obtain the asymptotic form of the distribution of the hitting time of the origin and that of the transition probability for the walk killed when it hits a finite set. The asymptotic forms obtained are valid uniformly in the natural domain of the space and time variables.
We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph is uniformly transient, and sublinearly in the expected size if the graph is uniformly recurrent with subexponential volume growth. This in particular answers a question asked by Benjamini, Kozma, Yadin and Yehudayoff (arXiv:0903.3179v1).
This work is motivated by the study of some two-dimensional random walks in random environment (RWRE) with transition probabilities independent of one coordinate of the walk. These are non-reversible models and can not be treated by electrical network techniques. The proof of the recurrence of such RWRE needs new estimates for quenched return probabilities of a one-dimensional recurrent RWRE. We obtained these estimates by constructing suitable valleys for the potential. They imply that k independent walkers in the same one-dimensional (recurrent) environment will meet in the origin infinitely…
We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, $a(x)$, of the random walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a criterion for $a(x)$ to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin.
We consider the extreme value statistics of centrally-biased random walks with asymptotically-zero drift in the ergodic regime. We fully characterize the asymptotic distribution of the maximum for this class of Markov chains lacking translational invariance, with a particular emphasis on the relation between the time scaling of the expected value of the maximum and the stationary distribution of the process.
We describe scaling limits of recurrent excited random walks (ERWs) on integers in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, delta, belongs to the interval [-1,1]. We show that if |delta|<1 then the diffusively scaled ERW under the averaged measure converges to a (delta,-delta)-perturbed Brownian motion. In the boundary case, |delta|=1, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens…
We consider a one-dimensional recurrent random walk in random environment (RWRE) when the environment is i.i.d. with a parametric, finitely supported distribution. Based on a single observation of the path, we provide a maximum likelihood estimation procedure of the parameters of the environment. Unlike most of the classical maximum likelihood approach, the limit of the criterion function is in general a nondegenerate random variable and convergence does not hold in probability. Not only the leading term but also the second order asymptotics is needed to fully identify the unknown parameter. W…
In this paper, we are interested in the asymptotic behaviour of the sequence of processes $(W_n(s,t))_{s,t\in[0,1]}$ with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(1_{\{ξ_{S_k}\leq s\}}-s\big) \end{equation*} where $(ξ_x, x\in\mathbb{Z}^d)$ is a sequence of independent random variables uniformly distributed on $[0,1]$ and $(S_n)_{n\in\mathbb N}$ is a random walk evolving in $\mathbb{Z}^d$, independent of the $ξ$'s. In Wendler (2016), the case where $(S_n)_{n\in\mathbb N}$ is a recurrent random walk in $\mathbb{Z}$ such that $(n^{-\frac 1α}S_n)_{n\geq 1}$ converges in distr…
We define a random walk on the set of primitive points of $\mathbb{Z}^d$. We prove that for walks generated by measures satisfying mild conditions these walks are recurrent in a strong sense. That is, we show that the associated Markov chains are positive recurrent and there exists a unique stationary measure for the random walk.
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…
We consider one infinite path of a Random Walk in Random Environment (RWRE, for short) in an unknown environment. This environment consists of either i.i.d.\ site or bond randomness. At each position the random walker stops and tells us the environment it sees at the point where it is, without telling us, where it is. These observations $χ'$ are spoiled by reading errors that occur with probability $p<1$. We show: If the RWRE is recurrent and satisfies the standard assumptions on such RWREs, then with probability one in the environment, the errors, and the random walk we are able reconstruct t…
Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding Markov chains (or 1D random walks) allow, in one transition, to reach for the N-th previous states, to remain in the state or reach for the immediately next state. The dual Markov chains allow, in one transition, to reach for the N-th next states, to remain in the state or reach for immediately previous state. T…
Simple random walks on various types of partially horizontally oriented regular lattices are considered. The horizontal orientations of the lattices can be of various types (deterministic or random) and depending on the nature of the orientation the asymptotic behaviour of the random walk is shown to be recurrent or transient. In particular, for randomly horizontally oriented lattices the random walk is almost surely transient.
Network growth models that embody principles such as preferential attachment and local attachment rules have received much attention over the last decade. Among various approaches, random walks have been leveraged to capture such principles. In this paper we consider the No Restart Random Walk (NRRW) model where a walker builds its graph (tree) while moving around. In particular, the walker takes s steps (a parameter) on the current graph. A new node with degree one is added to the graph and connected to the node currently occupied by the walker. The walker then resumes, taking another s steps…
In this paper, we investigate the properties of recurrent planar Markov random walks. More precisely, we study the set of recurrent points with the use of local limit theorems. The Nagaev-Guivarc'h spectral method provides several examples for which these local limit theorems are satisfied as soon as the (standard or non-standard) central limit theorem holds.
Let X_0=0, X_1, X_2, ..., be an aperiodic random walk generated by a sequence xi_1, xi_2, ..., of i.i.d. integer-valued random variables with common distribution p(.) having zero mean and finite variance. For an N-step trajectory X=(X_0,X_1,...,X_N) and a monotone convex function V: R^+ -> R^+ with V(0)=0, define V(X)= sum_{j=1}^{N-1} V(|X_j|). Further, let I_{N,+}^{a,b} be the set of all non-negative paths X compatible with the boundary conditions X_0=a, X_N=b. We discuss asymptotic properties of X in I_{N,+}^{a,b} w.r.t. the probability distribution P_{N}^{a,b}(X)= (Z_{N}^{a,b})^{-1} exp{-la…
We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.
Let $μ_1,... μ_k$ be $d$-dimensional probability measures in $\R^d$ with mean 0. At each step we choose one of the measures based on the history of the process and take a step according to that measure. We give conditions for transience of such processes and also construct examples of recurrent processes of this type. In particular, in dimension 3 we give the complete picture: every walk generated by two measures is transient and there exists a recurrent walk generated by three measures.
We show that the "twisted" planar random walk - which results by summing up stationary increments rotated by multiples of a fixed angle - is recurrent under diverse assumptions on the increment process. For example, if the increment process is alpha-mixing and of finite second moment, then the twisted random walk is recurrent for every angle fixed choice of the angle out of a set of full Lebesgue measure, no matter how slow the mixing coefficients decay.
We consider reversible random walks in random environment obtained from symmetric long--range jump rates on a random point process. We prove almost sure transience and recurrence results under suitable assumptions on the point process and the jump rate function. For recurrent models we obtain almost sure estimates on effective resistances in finite boxes. For transient models we construct explicit fluxes with finite energy on the associated electrical network.
We prove that the linearly edge reinforced random walk (LRRW) on any graph with bounded degrees is recurrent for sufficiently small initial weights. In contrast, we show that for non-amenable graphs the LRRW is transient for sufficiently large initial weights, thereby establishing a phase transition for the LRRW on non-amenable graphs. While we rely on the description of the LRRW as a mixture of Markov chains, the proof does not use the magic formula. We also derive analogous results for the vertex reinforced jump process.
We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16]. Here we study the heavy range: the number of edges visited at least $α$ times for some integer $α$. The asymptotic behavior of this process when $α$ is a power of the number of steps of the walk is given for all the recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a s…
The Tree Builder Random Walk is a special random walk that evolves on trees whose size increases with time, randomly and depending upon the walker. After every s steps of the walker, a random number of vertices are added to the tree and attached to the current position of the walker. These processes share similarities with other important classes of markovian and non-markovian random walks presenting a large variety of behaviors according to parameters specifications. We show that for a large and most significant class of tree builder random walks, the process is either null recurrent or trans…
In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is of the maximal order square root of n. In higher dimensions we call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely many (resp. finitely many) of the straight lines between two consecutive sites hit a given bounded set. The above estimate implies that three-dimensional random walks with independent components are polygonally transient. Similarly a directionally reinforced random walk on Z^3 in the…
Consider a walker performing a random walk in an i.i.d. random environment, and assume that the walker tells us at each time the environment it sees at its present location. Given this history of the transition probabilities seen from the walker - but not its trajectory - can we tell if the RWRE is recurrent or transient? Can we reconstruct the law of the environment? We show that in a one-dimensional environment, the law of the environment can be reconstructed, and we know in particular if the RWRE is recurrent or transient.
We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^α \vee 1$, and each time an edge is traversed, its weight is increased by $Δ$. It is known that the walk is recurrent if and only if $α\leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $α<1$ and $Δ>0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $Δ>0$ is much slower than $Δ=0$. In the critical case $α=1$, ou…
We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a …
Inspired by Benjamini et al (Ann. Inst. H. Poincaré Probab. Stat. 2010) and Windisch (Electron. J. Probab. 2010), we consider the entropy of the random walk range formed by a simple random walk on a discrete group. It is shown in this setting the existence of a quantity which we call the average entropy of the ranges. Some equivalent conditions for the vanishing of the average entropy of the ranges are given. Particularly, the average entropy of the ranges vanishes if and only if the random walk is recurrent or escaping to negative infinity without left jump. In order to characterize the recur…
We consider excited random walk (ERW) on $\mathbb{Z}$ in environments with identical stacks of infinitely many cookies at each site, subject to the constraint that the total drift per site $δ= \sum (2p_j - 1)$ is finite. Building on the methods of Kozma, Orenshtein, and Shinkar (arXiv:1311.7439), we show that ERW in finite-drift environments is recurrent when $|δ|<1$ and transient when $|δ|>1$. In the case $|δ|=1$ we prove that ERW is recurrent under mild assumptions on the environment. In addition, we show that ERW may be transient when $|δ|=1$, an interesting new behavior that was not presen…
Let $(M_{n},S_{n})_{n\ge 0}$ be a Markov random walk with positive recurrent driving chain $(M_{n})_{n\ge 0}$ having countable state space $\mathcal{S}$ and stationary distribution $π$. It is shown in this note that, if the dual sequence $({}^{\#}M_{n},{}^{\#}S_{n})_{n\ge 0}$ is positive divergent, i.e. ${}^{\#}S_{n}\to\infty$ a.s., then the strictly ascending ladder epochs $σ_{n}^{>}$ of $(M_{n},S_{n})_{n\ge 0}$ are a.s. finite and the ladder chain $(M_{σ_{n}^{>}})_{n\ge 0}$ is positive recurrent on some $\mathcal{S}^{>}\subset\mathcal{S}$. We also provide simple expressions for its stationar…
Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by rand…
We consider a recurrent random walk in random environment on a regular tree. Under suitable general assumptions upon the distribution of the environment, we show that the walk exhibits an unusual slow movement: the order of magnitude of the walk in the first $n$ steps is $(\log n)^3$.
We prove that any vertex-reinforced random walk on the integer lattice with non-decreasing reinforcement sequence $w$ satisfying $w(k) = o(k^α)$ for some $α< 1/2$ is recurrent. This improves on previous results of Volkov (2006) and Schapira (2012).
We consider the problem of the first passage time to the origin of a spatially non-homogeneous random walk with a position-dependent drift, known as the Gillis random walk, in the presence of resetting. The walk starts from an initial site $ x_0 $ and, with fixed probability $ r $, at each step may be relocated to a given site $ x_r $. From a general perspective, we first derive a series of results regarding the first and the second moment of the first hitting time distribution, valid for a wide class of processes, including random walks lacking the property of translational invariance; we the…
We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $μ$ and $ν$ are supercritical offspring distributions with means $\bar μ< \bar ν$ then every space-time set that is recurrent with respect to the offspring distribution $μ$ is also recurrent with …
In this paper we develop the idea of Lyons and gives a simple criterion for the recurrence and the transience. We also show that a wedge has the infinite collision property if and only if it is a recurrent graph.
The lonely branching random walks on ${\mathbb Z}^d$ is an interacting particle system where each particle moves as an independent random walk and undergoes critical binary branching when it is alone. We show that if the symmetrized walk is recurrent, lonely branching random walks die out locally. Furthermore, the same result holds if additional branching is allowed when the walk is not alone.
We introduce random walks in a sparse random environment on $\mathbb Z$ and investigate basic asymptotic properties of this model, such as recurrence-transience, asymptotic speed, and limit theorems in both the transient and recurrent regimes. The new model combines features of several existing models of random motion in random media and admits a transparent physical interpretation. More specifically, a random walk in a sparse random environment can be characterized as a "locally strong" perturbation of a simple random walk by a random potential induced by "rare impurities," which are randomly…
The joint distribution of value and local time for Brownian Motion has been reported by Borodin and Salminen. Its asymptotic behavior for recurrent random walk has been presented by Jain and Pruitt. Motivated by the need for queue size control during a pandemic (Hassin, Meilijson and Perlman), the current study presents closed form formulas for random walk and reflected random walk with $\pm 1$ increments, not necessarily fair.
We analyze random walk in the upper half of a three dimensional lattice which goes down whenever it encounters a new vertex, a.k.a. excited random walk. We show that it is recurrent with an expected number of returns of square-root log n.
