Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFunctional central limit theorem for random walks in random environment defined on regular trees
69
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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 the process is defined on $b$-regular trees, with $ b ge 4$, substantially improving the results of the first author (see Theorem 3 of Collevecchio (2006)).
rate research
Read More
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. When $d ge 3$ and the fluctuation of the environment is well moderated by the random walk, we prove a central limit theorem for the density of the population, together with upper bounds for the density of the most populated site and the replica overlap. We also discuss the phase transition of this model in connection with directed polymers in random environment.
We prove a quenched central limit theorem for random walks in i.i.d. weakly elliptic random environments in the ballistic regime. Such theorems have been proved recently by Rassoul-Agha and Seppalainen in [10] and Berger and Zeitouni in [2] under the assumption of large finite moments for the regeneration time. In this paper, with the extra $(T)_{gamma}$ condition of Sznitman we reduce the moment condition to ${Bbb E}(tau^2(ln tau)^{1+m})<+infty$ for $m>1+1/gamma$, which allows the inclusion of new non-uniformly elliptic examples such as Dirichlet random environments.
We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method of investigations based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.
We obtain Central Limit Theorems in Functional form for a class of time-inhomogeneous interacting random walks on the simplex of probability measures over a finite set. Due to a reinforcement mechanism, the increments of the walks are correlated, forcing their convergence to the same, possibly random, limit. Random walks of this form have been introduced in the context of urn models and in stochastic approximation. We also propose an application to opinion dynamics in a random network evolving via preferential attachment. We study, in particular, random walks interacting through a mean-field rule and compare the rate they converge to their limit with the rate of synchronization, i.e. the rate at which their mutual distances converge to zero. Under certain conditions, synchronization is faster than convergence.
We consider branching random walks in $d$-dimensional integer lattice with time-space i.i.d. offspring distributions. This model is known to exhibit a phase transition: If $d ge 3$ and the environment is not too random, then, the total population grows as fast as its expectation with strictly positive probability. If,on the other hand, $d le 2$, or the environment is ``random enough, then the total population grows strictly slower than its expectation almost surely. We show the equivalence between the slow population growth and a natural localization property in terms of replica overlap. We also prove a certain stronger localization property, whenever the total population grows strictly slower than its expectation almost surely.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
