Transient Random Walks Research Articles

Some asymptotic results for transient random walks

We consider a real-valued random walkSwhich drifts to –∞ and is such thatE(expθS1) < ∞ for someθ> 0, but for which Cramér's condition fails. We investigate the asymptotic tail behaviour of the distributions of the all time maximum, the upwards and downwards first passage times and the last passage times. As an application, we obtain new limit theorems for certain conditional laws.

The limit theorems for the single point range of strongly transient random walks

The limit theorems for the single point range of strongly transient random walks

Entropic repulsion of the lattice free field

Consider the massless free field on thed-dimensional lattice ℤd,d≧3; that is the centered Gaussian field on with covariances given by the Green function of the simple random walk on ℤd. We show that the probability, that all the spins are positive in a box of volumeN d, decays exponentially at a rate of orderN d−2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and α-stable law.

Large loss networks

A general loss network is considered in the limit as the arrival rates and link capacities become large with their ratio held fixed. We show that the network obeys a functional law of large numbers (along a subsequence) and that the free circuit process acts as a control for the network. The network exhibits a separation of time-scales, with the free circuit process operating on the fast time-scale as a random walk on Z + J , and this leads to an interesting conjecture for transient random walks. The techniques used to prove the results are of independent interest and can be applied to a wide range of models in which a similar separation of time-scale occurs, or in which the transition rates of the process undergo a discontinuity at or near a boundary. Finally, we give examples and show that commonly employed fixed point approximations are not valid in this limit.

Large Deviation Theorems for the First Time of Crossing an Increasing Level in a Transient Random Walk

Large Deviation Theorems for the First Time of Crossing an Increasing Level in a Transient Random Walk

Algebraic $L^2$ Decay of Attractive Critical Processes on the Lattice

We consider a special class of attractive critical processes based on the transition function of a transient random walk on $\mathbb{Z}^d$. These processes have infinitely many invariant distributions and no spectral gap. The exponential $L^2$ decay is replaced by an algebraic $L^2$ decay. The paper shows the dependence of this algebraic rate in terms of the dimension of the lattice and the locality of the functions under consideration. The theory is illustrated by several examples dealing with locally interacting diffusion processes and independent random walks.

Isoperimetric Inequalities and Transient Random Walks on Graphs

recurrent. We show that any graph satisfying an isoperimetric inequality only slightly stronger than that of Z2 is transient. More precisely, if f(k) denotes the smallest number of vertices in the boundary of a connected subgraph with k vertices, then the graph is transient if the infinite sum E f(k)-2 converges. This can be applied to parabolicity versus hyperbolicity of surfaces. 1. Introduction. Let G be a connected graph which is locally finite, that is, all vertices have finite degree. We consider a random walk starting at a vertex v, say, such that at any vertex u, the walk proceeds to a neighbour with probability 1/d(u), where d(u) is the degree (i.e., the number of neighbours) of u. The graph G is recurrent if we revisit v with probability 1. Otherwise G is transient. It is well known that the three-dimensional grid Z3 is transient while Z2 is recurrent. More generally, Nash-Williams [13] (see also [4, 10]) proved that any graph with smaller growth rate than Z2 is recurrent. Lyons [10] showed that certain subgraphs of grids are transient provided they grow just a little faster than Z2. Other results, in terms of isoperimetric inequalities, supporting the statement that Z2 is, in a sense, an extreme recurrent graph, can be derived from work of Fernandez [5], Grigor'yan [7] and Varopoulos [15]. (Varopoulos [16] used results of Gromov [8] to characterize completely the recurrent Cayley graphs.) We shall carry these results further. If V is a vertex set in G, then aV will denote the boundary of V, that is, the set of vertices of V having neighbours outside of V. Let f be a nondecreasing positive real function defined on the natural numbers. We say that G satisfies an f-isoperimetric inequality if there exists a constant c > 0 such that, for each finite vertex set V of G,

Transient Random Walks on Graphs and Metric Spaces with Applications to Hyperbolic Surfaces

We introduce an (r, /?)-net (0 < 2r < R) of a metric space M as a maximal graph whose vertices are elements in M of pairwise distance at least r such that any two vertices of distance at most R are adjacent. We show that, for a large class of metric spaces, including many Riemannian manifolds, the property of transience of a net and the property of the net carrying a non-constant harmonic function of bounded energy is independent of the choice of the net. We give a new necessary and sufficient condition for a graph with bounded degrees and satisfying an isoperimetric inequality to have no non-constant harmonic functions. For this purpose we develop equivalent analytic conditions for graphs satisfying an isoperimetric inequality. Some of these results have been discovered recently by others in more general settings, but our treatment here is specific and self-contained. We use graph transience to prove that Scherk's surface is hyperbolic, a problem posed by Osserman in 1965.

Harmonic functions on cartesian products of trees with finite graphs

Let G be a graph which is the Cartesian product of an infinite, locally finite tree T and a finite, connected graph A . On G , consider a stochastic transition operator P giving rise to a transient random walk and such that positive transitions occur only along the edges of G . We construct a matrix-valued kernel on T , which extends naturally in the second variable to the space of ends Ω of T . This kernel is used to derive a unique integral representation over Ω of all—not necessarily positive—functions on G which are harmonic with respect to P . We explain the relation with the Martin boundary and the positive harmonic functions and, as a particular case, we show what happens when A arises from a finite abelian group and P is compatible with the structure of A .

On a Maximum of a transient random walk in random environment

On a Maximum of a transient random walk in random environment

Harmonic functions and ends of graphs

In this note we discuss extensions of results [5], where transient random walks are considered, whose transition matrix is compatible with a tree-structure of the underlying discrete state space. Notation is generally as in [5], with the exception that instead of a tree T we consider an arbitrary graph Γ, which is locally finite, connected, and whose vertex set S is infinite. The edge set E is unoriented, there are no multiple edges. We consider a transition operator P, given by a stochastic matrix (p(u, v))u, v ∈ S, which gives rise to a transient Markov chain (“random walk”), related to the graph structure by:(i) if p(u, v)&gt;0 then [u, v] ∈ E,(ii) there is an M&gt;0 such that p(k)(u, v)&gt;0 for some k=k(u, v)≦M whenever [u, v]∈E (“uniform irreducibility”).

Martin Boundaries of Random Walks: Ends of Trees and Groups

Consider a transient random walk ${X_n}$ on an infinite tree $T$ whose nonzero transition probabilities are bounded below. Suppose that ${X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, ${X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $T$.) The Martin boundary of ${X_n}$ is shown to coincide with the space $\Omega$ of all ends of $T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $\Omega$ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.

Martin boundaries of random walks: ends of trees and groups

Consider a transient random walk ${X_n}$ on an infinite tree $T$ whose nonzero transition probabilities are bounded below. Suppose that ${X_n}$ is uniformly irreducible and has bounded step-length. (Alternatively, ${X_n}$ can be regarded as a random walk on a graph whose metric is equivalent to the metric of $T$.) The Martin boundary of ${X_n}$ is shown to coincide with the space $\Omega$ of all ends of $T$ (or, equivalently, of the graph). This yields a boundary representation theorem on $\Omega$ for all positive eigenfunctions of the transition operator, and a nontangential Fatou theorem which describes their boundary behavior. These results apply, in particular, to finitely supported random walks on groups whose Cayley graphs admit a uniformly spanning tree. A class of groups of this type is constructed.

𝐿²-cohomology of noncompact surfaces

This paper is motivated by the question of whether nonzero L 2 {L^2} -harmonic differentials exist on coverings of a Riemann surface of genus ⩾ 2 \geqslant 2 . Our approach will be via an analogue of the de Rham theorem. Some results concerning the invariance of L 2 {L^2} -homology and the intersection number of L 2 {L^2} -cycles are demonstrated. A growth estimate for triangulations of planar coverings of the two-holed torus is derived. Finally, the equivalence between the existence of L 2 {L^2} -harmonic one-cycles and the transience of random walks on a planar surface is shown.

Sur la theorie du renouvellement pour les groupes non abfliens

Assuming that their laws are spread-out, we extend the renewal theorem to transient random walks on several classes of non-abelian groups.

Random walk point processes

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.

Multiple Points of Transient Random Walks

Multiple Points of Transient Random Walks

Random walk point processes

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.

Multiple points of transient random walks

We determine the asymptotic behavior of the expected numbers of points visited exactly $j$ times and at least $j$ times in the first $n$ steps of a transient random walk on a discrete Abelian group. We prove that the strong law of large numbers holds for these multiple point ranges.

The central limit theorem for the range of transient random walk

The central limit theorem for the range of transient random walk