Random walks in a strongly sparse random environment

The integer points (sites) of the real line are marked by the positions of a standard random walk with positive integer jumps. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity and assuming additionally that the distribution tail of the jumps is regularly varying at infinity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1∕2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2019) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable.     

Lagging and leading coupled continuous time random walks, renewal times and their joint limits

Subordinating a random walk to a renewal process yields a continuous time random walk (CTRW), which models diffusion and anomalous diffusion. Transition densities of scaling limits of power law CTRWs have been shown to solve fractional Fokker-Planck equations. We consider limits of CTRWs which arise when both waiting times and jumps are taken from an infinitesimal triangular array. Two different limit processes are identified when waiting times precede jumps or follow jumps, respectively, together with two limit processes corresponding to the renewal times. We calculate the joint law of all four limit processes evaluated at a fixed time t.     

A note on multitype branching process with bounded immigration in random environment

In this paper, we study the total number of progeny, W, before regenerating of multitype branching process with immigration in random environment. We show that the tail probability of |W| is of order t-κ as t→∞, with κ some constant. As an application, we prove a stable law for (L-1) random walk in random environment, generalizing the stable law for the nearest random walk in random environment (see "Kesten, Kozlov, Spitzer: A limit law for random walk in a random environment. Compositio Math., 30, 145-168 (1975)").     

The hydrodynamic limit for a system with interactions prescribed by Ginzburg-Landau type random Hamiltonian

Summary As a microscopic model we consider a system of interacting continuum like spin field overR ^d . Its evolution law is determined by the Ginzburg-Landau type random Hamiltonian and the total spin of the system is preserved by this evolution. We show that the spin field converges, under the hydrodynamic space-time scalling, to a deterministic limit which is a solution of a certain nonlinear diffusion equation. This equation describes the time evolution of the macroscopic field. The hydrodynamic scaling has an effect of the homogenization on the system at the same time.     

Invariance principle for nonhomogeneous random walks on the grid ℤ1

A nonhomogeneous random walk on the grid ℤ¹ with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables and Brownian motion. Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 459–472, September, 1999.     

Scaling limit of local time of Sinai’s random walk

We prove that the local times of a sequence of Sinai’s random walks converge to those of Brox’s diffusion by proper scaling. Our proof is based on the intrinsic branching structure of the random walk and the convergence of the branching processes in random environment.     

Patterns in random permutations avoiding the pattern 321

We consider a random permutation drawn from the set of 321 ‐avoiding permutations of length n and show that the number of occurrences of another pattern σ has a limit distribution, after scaling by n^{m + ?} where m is the length of σ and ? is the number of blocks in it. The limit is not normal, and can be expressed as a functional of a Brownian excursion.     

Triangular array limits for continuous time random walks

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based on triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space–time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed.     

Weak interaction limits for one-dimensional random polymers

 In this paper we present a new and flexible method to show that, in one dimension, various self-repellent random walks converge to self-repellent Brownian motion in the limit of weak interaction after appropriate space-time scaling. Our method is based on cutting the path into pieces of an appropriately scaled length, controlling the interaction between the different pieces, and applying an invariance principle to the single pieces. In this way, we show that the self-repellent random walk large deviation rate function for the empirical drift of the path converges to the self-repellent Brownian motion large deviation rate function after appropriate scaling with the interaction parameters. The method is considerably simpler than the approach followed in our earlier work, which was based on functional analytic arguments applied to variational representations and only worked in a very limited number of situations. We consider two examples of a weak interaction limit: (1) vanishing self-repellence, (2) diverging step variance. In example (1), we recover our earlier scaling results for simple random walk with vanishing self-repellence and show how these can be extended to random walk with steps that have zero mean and a finite exponential moment. Moreover, we show that these scaling results are stable against adding self-attraction, provided the self-repellence dominates. In example (2), we prove a conjecture by Aldous for the scaling of self-avoiding walk with diverging step variance. Moreover, we consider self-avoiding walk on a two-dimensional horizontal strip such that the steps in the vertical direction are uniform over the width of the strip and find the scaling as the width tends to infinity. Received: 6 March 2002 / Revised version: 11 October 2002 / Published online: 21 February 2003 Mathematics Subject Classification (2000): 60F05, 60F10, 60J55, 82D60 Key words or phrases: Self-repellent random walk and Brownian motion – Invariance principles – Large deviations – Scaling limits – Universality     

Scenery reconstruction with branching random walk

We study the problem of scenery reconstruction in arbitrary dimension using observations registered in boxes of size k (for k fixed), seen along a branching random walk. We prove that, using a large enough k for almost all the realizations of the branching random walk, almost all sceneries can be reconstructed up to equivalence.     

Hitting time of large subsets of the hypercube

We study the simple random walk on the n‐dimensional hypercube, in particular its hitting times of large (possibly random) sets. We give simple conditions on these sets ensuring that the properly rescaled hitting time is asymptotically exponentially distributed, uniformly in the starting position of the walk. These conditions are then verified for percolation clouds with densities that are much smaller than (n log n)^‐1. A main motivation behind this article is the study of the so‐called aging phenomenon in the Random Energy Model, the simplest model of a mean‐field spin glass. Our results allow us to prove aging in the REM for all temperatures, thereby extending earlier results to their optimal temperature domain. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Thoughts on modeling complexity

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This simple model has been applied to all manner of phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model beyond that of mimicking simple statistics to include long‐time memory in the dynamics of complex phenomena. We show that complexity can give rise to fractional‐difference stochastic processes whose continuum limit is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. Furthermore, the index of the inverse power‐law spectrum in many complex processes can be related to the fractional derivative index in the fractional Langevin equation. This fractional stochastic model suggests that a scaling process guides the dynamics of many complex phenomena. The alternative to the fractional Langevin equation is a fractional diffusion equation describing the evolution of the probability density for certain kinds of anomalous diffusion. © 2006 Wiley Periodicals, Inc. Complexity 11: 33–43, 2006     

Scaling limits for internal aggregation models with multiple sources

We study the scaling limits of three different aggregation models on ℤ ^d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝ ^d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.     

Hitting time of a half-line by two-dimensional random walk

We consider the probability that a two-dimensional random walk starting from the origin never returns to the half-line {(x₁,x₂)|x₁0,x₂=0} before time n. It is proved that for aperiodic random walk with mean zero and finite 2+(>2)-th absolute moment, this probability times n^1/4 converges to some positive constant c^* as . We show that c^* is expressed by using the characteristic function of the increment of the random walk. For the simple random walk, this expression gives Mathematics Subject Classification (2000):60G50, 60E10     

The mean number of sites visited by a pinned random walk

This paper concerns the number Z _n of sites visited up to time n by a random walk S _n having zero mean and moving on the d-dimensional square lattice Z ^d . Asymptotic evaluation of the conditional expectation of Z _n given that S ₀ = 0 and S _n = x is carried out under 2 + δ moment conditions (0 ≤ δ ≤ 2) in the cases d = 2, 3. It gives an explicit form of the leading term and reasonable estimates of the remainder term (depending on δ) valid uniformly in each parabolic region of (x, n). In the case x = 0 the problem has been studied for the simple random walk and its analogue for Brownian motion; the estimates obtained here are finer than or comparable to those found in previous works. Supported in part by Monbukagakusho grand-in-aid no. 15540109.     

Macroscaling Limit Theorems for Filtered Spatiotemporal Random Fields

This article addresses the problem of defining a general scaling setting in which Gaussian and non-Gaussian limit distributions of linear random fields can be obtained. The linear random fields considered are defined by the convolution of a Green kernel, satisfying suitable scaling conditions, with a non-linear transformation of a Gaussian centered homogeneous random field. The results derived cover the weak-dependence and strong-dependence cases for such Gaussian random fields. Extension to more general random initial conditions defined, for example, in terms of non-linear transformations of χ²-random fields, is also discussed. For an example, we consider the random fractional diffusion equation. The vectorial version of the limit theorems derived is also formulated, including the limit distribution of the parabolically rescaled solution to the Burgers equation in the cases of weakly and strongly dependent initial potentials.     

Random walks in random environments on metric groups

Random walks in random environments on countable metric groups with bounded jumps of the walking particle are considered. The transition probabilities of such a random walk from a pointx εG (whereG is the group in question) are described by a vectorp(x) ε ℝ^|W| (whereW ⊏G is fixed and |W|<∞). The set {p(x),x εG} is assumed to consist of independent identically distributed random vectors. A sufficient condition for this random walk to be transient is found. As an example, the groups ℤ ^d , free groups, and the free product of finitely many cyclic groups of second order are considered. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 129–135, January, 2000.     

Diffusion of color in the simple exclusion process

We prove a diffusion scaling limit for the macroscopic densities of colored particles performing the simply excluded random walk, and relate this to the limiting behavior of a test particle in equilibrium.     

Scaling limit theorem for transient random walk in random environment

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.     

Convergence to fractional kinetics for random walks associated with unbounded conductances

We consider a random walk among unbounded random conductances whose distribution has infinite expectation and polynomial tail. We prove that the scaling limit of this process is a Fractional-Kinetics process??that is the time change of a d-dimensional Brownian motion by the inverse of an independent ??-stable subordinator. We further show that the same process appears in the scaling limit of the non-symmetric Bouchaud??s trap model.