Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userRandom walk on random walks: low densities
92
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We consider a random walker in a dynamic random environment given by a system of independent simple symmetric random walks. We obtain ballisticity results under two types of perturbations: low particle density, and strong local drift on particles. Surprisingly, the random walker may behave very differently depending on whether the underlying environment particles perform lazy or non-lazy random walks, which is related to a notion of permeability of the system. We also provide a strong law of large numbers, a functional central limit theorem and large deviation bounds under an ellipticity condition.
rate research
Read More
We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].
We study random walks on the giant component of the ErdH{o}s-Renyi random graph ${cal G}(n,p)$ where $p=lambda/n$ for $lambda>1$ fixed. The mixing time from a worst starting point was shown by Fountoulakis and Reed, and independently by Benjamini, Kozma and Wormald, to have order $log^2 n$. We prove that starting from a uniform vertex (equivalently, from a fixed vertex conditioned to belong to the giant) both accelerates mixing to $O(log n)$ and concentrates it (the cutoff phenomenon occurs): the typical mixing is at $( u {bf d})^{-1}log n pm (log n)^{1/2+o(1)}$, where $ u$ and ${bf d}$ are the speed of random walk and dimension of harmonic measure on a ${rm Poisson}(lambda)$-Galton-Watson tree. Analogous results are given for graphs with prescribed degree sequences, where cutoff is shown both for the simple and for the non-backtracking random walk.
We study the limiting occupation density process for a large number of critical and driftless branching random walks. We show that the rescaled occupation densities of $lfloor sNrfloor$ branching random walks, viewed as a function-valued, increasing process ${g_{s}^{N}}_{sge 0}$, converges weakly to a pure jump process in the Skorohod space $mathbb D([0, +infty), mathcal C_{0}(mathbb R))$, as $Ntoinfty$. Moreover, the jumps of the limiting process consist of i.i.d. copies of an Integrated super-Brownian Excursion (ISE) density, rescaled and weighted by the jump sizes in a real-valued stable-1/2 subordinator.
We study models of continuous time, symmetric, $Z^d$-valued random walks in random environments. One of our aims is to derive estimates on the decay of transition probabilities in a case where a uniform ellipticity assumption is absent. We consider the case of independent conductances with a polynomial tail near 0, and obtain precise asymptotics for the annealed return probability and convergence times for the random walk confined to a finite box.
We introduce a new type of random walk where the definition of edge reinforcement is very different from the one in the reinforced random walk models studied so far, and investigate its basic properties, such as null/positive recurrence, transience, and speed. Two basic cases will be dubbed impatient andageing random walks.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
