Let $b$ be an integer greater than 1 and let $W^{\ee}=(W^{\ee}_n; n\geq 0)$ be a random walk on the $b$-ary rooted tree $\U_b$, starting at the root, going up (resp. down) with probability $1/2+ε$ (resp. $1/2 -ε$), $ε\in (0, 1/2)$, and choosing direction $i\in \{1, ..., b\}$ when going up with probability $a_i$. Here $å=(a_1, ..., a_b)$ stands for some non-degenerated fixed set of weights. We consider the range $\{W^{\ee}_n ; n\geq 0 \}$ that is a subtree of $\U_b $. It corresponds to a unique random rooted ordered tree that we denote by $τ_ε$. We rescale the edges of $τ_ε$ by a factor $\ee $ and we let $\ee$ go to 0: we prove that correlations due to frequent backtracking of the random walk only give rise to a deterministic phenomenon taken into account by a positive factor $γ(å)$. More precisely, we prove that $τ_ε$ converges to a continuum random tree encoded by two independent Brownian motions with drift conditioned to stay positive and scaled in time by $γ(å)$. We actually state the result in the more general case of a random walk on a tree with an infinite number of branches at each node ($b=\infty$) and for a general set of weights $å=(a_n, n\geq 0)$.
In this article it is shown that the Brownian motion on the continuum random tree is the scaling limit of the simple random walks on any family of discrete $n$-vertex ordered graph trees whose search-depth functions converge to the Brownian excursion as $n\to\infty$. We prove both a quenched version (for typical realisations of the trees) and an annealed version (averaged over all realisations of the trees) of our main result. The assumptions of the article cover the important example of simple random walks on the trees generated by the Galton-Watson branching process, conditioned on the total…
The Pólya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the Pólya process. We deal with a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard Pólya process. In one of the scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn," but a broader int…
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 define and characterise regular sequences in affine buildings, thereby giving the "$p$-adic analogue" of the fundamental work of Kaimanovich. As applications we prove limit theorems for random walks on affine buildings and their automorphism groups.
We prove existence of asymptotic entropy of random walks on regular languages over a finite alphabet and we give formulas for it. Furthermore, we show that the entropy varies real-analytically in terms of probability measures of constant support, which describe the random walk. This setting applies, in particular, to random walks on virtually free groups.
Let F be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F. An application to ruin probabilities is developed.
The food web is a directed graph in which nodes label species and directed links represent the predation between species. The cascade model generates random food webs. The continuum cascade model is a Poisson approximation of the cascade model. The recursion to obtain the probability distribution of the longest chain length generated by the continuum cascade model has the solution with traveling wave. We consider a branching random walk to study the asymptotic probability on the position of wave front.
We study a model of multi-excited random walk on a regular tree which generalizes the models of the once excited random walk and the digging random walk introduced by Volkov (2003). We show the existence of a phase transition of the recurrence/transience property of the walk. In particular, we prove that the asymptotic behavior of the walk depends on the order of the excitations, which contrasts with the one dimensional setting studied by Zerner (2005). We also consider the limiting speed of the walk in the transient regime and conjecture that it is not a monotonic function of the environment.
Random walks on a group $G$ model many natural phenomena. A random walk is defined by a probability measure $p$ on $G$. We are interested in asymptotic properties of the random walks and in particular in the linear drift and the asymptotic entropy. If the geometry of the group is rich, then these numbers are both positive and the way of dependence on $p$ is itself a property of $G$. In this note, we review recent results about the regularity of the drift and the entropy for free groups, free products and hyperbolic groups.
In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.
We prove that the speed of $λ$-biased random walks on a supercritical Galton-Watson tree without leaves is differentiable when $λ\in(0,1)$, and give an expression of the derivative using a certain 2-dimensional Gaussian random variable. The proof heavily uses the renewal structure of Galton-Watson trees that was introduced by Lyons-Pemantle-Peres.
Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain an explicit representation of the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.
We consider the simple random walk on a random d-regular graph with n vertices, and investigate percolative properties of the set of vertices not visited by the walk until time un, where u > 0 is a fixed positive parameter. It was shown in [arXiv:1012.5117] that this so-called vacant set exhibits a phase transition at u = u*: there is a giant component if u < u* and only small components when u > u*. In this paper we show the existence of a critical window of size n^(-1/3) around u*. In this window the size of the largest cluster is of order n^(2/3).
We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural …
Let $T$ be the regular tree in which every vertex has exactly $d\ge 3$ neighbours. Run a branching random walk on $T$, in which at each time step every particle gives birth to a random number of children with mean $d$ and finite variance, and each of these children moves independently to a uniformly chosen neighbour of its parent. We show that, starting with one particle at some vertex $0$ and conditionally on survival of the process, the time it takes for every vertex within distance $r$ of $0$ to be hit by a particle of the branching random walk is almost surely $r + \frac{2}{\log(3/2)}\log\…
In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the $n$-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the $n$-th generation.
We study Random Walks in an i.i.d. Random Environment (RWRE) defined on $b$-regular trees. We prove a functional central limit theorem (FCLT) for transient processes, under a moment condition on the environment. We emphasize that we make no uniform ellipticity assumptions. Our approach relies on regenerative levels, i.e. levels that are visited exactly once. On the way, we prove that the distance between consecutive regenerative levels have a geometrically decaying tail. In the second part of this paper, we apply our results to Linearly Edge-Reinforced Random Walk (LERRW) to prove FCLT when th…
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…
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…
Let $\{\mm_n, n=0,1,...\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,...$ let $W_n$ be the moment generating function of $\mm_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, $\su |W-W_n|$, $\su |W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that $\rP\{W_1>x\}$ regularly varies at $\infty$ it is proved that $\rP\{AW_n>x\}$ regularly varies at $\infty$ with the same exponent. All the proofs given …
We determine the walk dimension of the Sierpiński gasket without using diffusion. We construct non-local regular Dirichlet forms on the Sierpiński gasket from regular Dirichlet forms on the Sierpiński graph whose suitable boundary is the Sierpiński gasket.
We introduce a discrete-time random walk model on a one-dimensional lattice with a nonconstant sojourn time and prove that the discrete density converges to a solution of a continuum diffusion equation. Our random walk model is not Markovian due to the heterogeneity in the sojourn time, in contrast to a random walk model with a nonconstant walk length. We derive a Markovian process by choosing appropriate subindexes of the time-space grid points, and then show the convergence of its discrete density through the parabolic-scale limit. We also find the Green's function of the continuum diffusion…
We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u<u*, then the lar…
We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main resu…
For a continuous-time catalytic branching random walk (CBRW) on Z, with an arbitrary finite number of catalysts, we study the asymptotic behavior of position of the rightmost particle when time tends to infinity. The mild requirements include the regular variation of the jump distribution tail for underlying random walk and the well-known L log L condition for the offspring numbers. In our classification, given in the previous paper, the analysis refers to supercritical CBRW. The principle result demonstrates that, after a proper normalization, the maximum of CBRW converges in distribution to …
Using renewal times and Girsanov's transform, we prove that the speed of the excited random walk is infinitely differentiable with respect to the bias parameter in $(0,1)$ for the dimension $d\ge 2$. At the critical point $0$, using a special method, we also prove that the speed is differentiable and the derivative is positive for every dimension $2\leq d\neq 3.$ However, this is not enough to imply that the speed is increasing in a neighborhood of $0.$ It still remains to prove the derivative is continuous at $0$. Moreover, this paper gives some results of monotonicity for $m-$excited random …
We extend classical results by A. V. Nagaev [Izv. Akad. Nauk UzSSR Ser. Fiz.--Mat. Nauk 6 (1969) 17--22, Theory Probab. Appl. 14 (1969) 51--64, 193--208] on large deviations for sums of i.i.d. regularly varying random variables to partial sum processes of i.i.d. regularly varying vectors. The results are stated in terms of a heavy-tailed large deviation principle on the space of càdlàg functions. We illustrate how these results can be applied to functionals of the partial sum process, including ruin probabilities for multivariate random walks and long strange segments. These results make preci…
We establish some relations between the spectra of simple and non-backtracking random walks on non-regular graphs, generalizing some well-known facts for regular graphs. Our two main results are 1) a quantitative relation between the mixing rates of the simple random walk and of the non-backtracking random walk 2) a variant of the "Ihara determinant formula" which expresses the characteristic polynomial of the adjacency matrix, or of the laplacian, as the determinant of a certain non-backtracking random walk with holomorphic weights.
A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.
In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to supercritical percolation clusters, graph trees converging to the continuum random tree and the homogenisation problem for nested fractals. A subsequential local limit theorem for the simple random walks on generalised Sierpinski carpet graphs is also presented.
We consider the random walk loop soup on the discrete half-plane corresponding to a central charge c in (0, 1]. We look at the clusters of discrete loops and show that the scaling limit of the outer boundaries of outermost clusters is the CLE(kappa) loop ensemble, with the same relation between kappa and c as in the continuum Brownian setting.
It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $d/ ((d-2)\log (d-1))\log n$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on the infinite $d$-regular tree. If one can map another infinite transitive graph onto $G_n$, then we can improve the strategy by using a comparison with the random walk on this transitive graph (instead of that of the regular tree), and we obtain a lower bound of the form $1/h \log n$, where $h$ is the entropy rate associated with the …
We consider biased random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. Axelsson-Fisk and Häggström established for this model a phase transition for the asymptotic linear speed $\overline{\mathrm{v}}$ of the walk. Namely, there exists some critical value $λ_{\mathrm{c}}>0$ such that $\overline{\mathrm{v}}>0$ if $λ\in (0,λ_{\mathrm{c}})$ and $\overline{\mathrm{v}}=0$ if $λ>λ_{\mathrm{c}}$. We show that the speed $\overline{\mathrm{v}}$ is continuous in $λ$ on the interval $(0,λ_{\mathrm{c}})$ and differentiable on $(0,λ_{\mathrm{c}}/2)$. Mo…
Let $\mm_n, n=0,1,...$ be the supercritical branching random walk, in which the number of direct descendants of one individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mm_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than -1. The paper presents sufficient conditions of the almost sure convergence of the series $\sum_{n=1}^{\infty}a(n)(W-W_n)$. Also we establish a criterion of finiteness of $\me W\log^+ W a(\log^+W)$ and $\me \log^+|\zi| a(…
We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $e_1-$exchangeable cookie environment and prove the law of large numbers for this case.
We address the theory of records for integrated random walks with finite variance. The long-time continuum limit of these walks is a non-Markov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated…
Symmetric heavily tailed random walks on $Z^d, d\geq 1,$ are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in $x,t, |x|+t\to\infty,$) asymptotic behavior of the transition probability $p(t,0,x)$ is obtained. The examples indicate that the regularity conditions are essential.
Benjamini,Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d. This is a continuum of random walks indexed by a parameter t. They proved that for d=3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d > 4 there almost surely do not exist such t. Hoffman showed that for d=2 there almost surely exists t such that the random walk at time t visits the origin only finitely many times. We refine the results of Hoffman for dynamical random walk on Z^2, showing that with probability one there are times when the or…
We study the asymptotic properties of nearest-neighbor random walks in 1d random environment under the influence of an external field of intensity $λ\in\mathbb{R}$. For ergodic shift-invariant environments, we show that the limiting velocity $v(λ)$ is always increasing and that it is everywhere analytic except at most in two points $λ_-$ and $λ_+$. When $λ_-$ and $λ_+$ are distinct, $v(λ)$ might fail to be continuous. We refine the assumptions in \cite{Z} for having a recentered CLT with diffusivity $σ^2(λ)$ and give explicit conditions for $σ^2(λ)$ to be analytic. For the random conductance m…
We propose a class of strongly efficient rare event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on the heavy-tailed sample path large deviations result recently established in Rhee, Blanchet, and Zwart (2016). The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are "universal" in the…
A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field.
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
Let $(G_n)_{n=1}^{\infty}$ be a sequence of finite graphs, and let Y_t be the length of a loop-erased random walk on G_n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G_n is the d-dimensional torus of size-length n for $d \geq 4$, the process $(Y_t)_{t=0}^{\infty}$, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Bro…
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.
