A Quenched Central Limit Theorem for Reversible Random Walks in a Random Environment on Z

Abstract

The main aim of this paper is to prove the quenched central limit theorem for reversible random walks in a stationary random environment on Z without having the integrability condition on the conductance and without using any martingale. The method shown here is particularly simple and was introduced by Depauw and Derrien [3]. More precisely, for a given realization ω of the environment, we consider the Poisson equation (Pω - I)g = f, and then use the pointwise ergodic theorem in [8] to treat the limit of solutions and then the central limit theorem will be established by the convergence of moments. In particular, there is an analogue to a Markov process with discrete space and the diffusion in a stationary random environment.

Giới hạn phương sai cho bước đi ngẫu nhiên trong không gian aZ

“…Một công cụ rất hữu hiệu trong nghiên cứu kỳ vọng của biến ngẫu nhiên đó là sử dụng toán tử Markov cùng với tính chất của kỳ vọng có điều kiện. Phương pháp này cũng đã được nghiên bước đi ngẫu nhiên trong môi trường ngẫu nhiên trong các bài báo của Depauw and Derrien (2009) và Lam (2014). Ta xét phương trình hàm có dạng…”

The stepping stone model in a random environment and the effect of local heterogneities on isolation by distance patterns

“…The corresponding central limit theorem was proved for the random walk among random conductances in [Lam14] (see also [Lam12]). Although our case doesn't take the form of a random walk among random conductances, the proof in [Lam12, Chapter 4] only requires the reversibility of the random walk, and can be applied by replacing the conductivity with N (ω)T 1 N (ω) and the reversible measure with N (ω)N 3 (ω).…”

“…It turns out that these random walks admit a reversible measure on Z. This allows us to prove an invariance principle for the trajectory of each of these random walks [Koz85,DMFGW89,Lam14,Der15]. To characterize the coalescence time of a pair of lineages, we introduce an auxiliary process which records the sequence of demes where the two lineages meet, and we show that this process also admits a reversible measure (which happens to be the square of the previous one, see Proposition 3.3).…”