A Correlation Inequality for Tree-Indexed Markov Chains

Abstract

Let T be a tree, i.e. an infinite, locally finite graph without loops or cycles. One vertex of T, designated 0, is called the root; we assume all other vertices have degree at least 2. Attach a random variable S _σ to each vertex σ of T as follows. Take S ₀ = 0 and for σ ≠ 0 choose S _σ randomly, with equal probabilities, from $\left\{ {{S_{{\tilde{\sigma }}}} - 1,{S_{{\tilde{\sigma }}}} + 1} \right\}$ where σ is the “predecessor” of σ in T (see §2 for precise definitions). We call the process $\left\{ {{S_{\sigma }}:\sigma \in T} \right\}$ a T-walk on Z; note that taking T = {0,1,2,…} with consecutive integers connected, we recover the (ordinary) simple random walk on Z. Allowing richer trees T we may observe quite different asymptotic behaviour. One can study this behaviour either by considering the levels $\left\{ {\sigma :\left| \sigma \right| = n} \right\}$ = T _n of T (|σ| is the distance from 0 to σ) or by observing the rays (infinite non-self-intersecting paths) in T. The first approach, which we adopt here, was initiated by Joffe and Moncayo [JM] who gave conditions for asymptotic normality of the empirical measures determined by $\left\{ {{S_{\sigma }}:\left| \sigma \right| = n} \right\}$. The second approach is used in [E], [LP] and [BP1]. To elucidate both approaches, we quote a theorem which relates three notions of “speed” for a T-walk, to dimensional properties of the boundary ∂T of T (∂T is the collection of rays in T, emanating from 0). Equip ∂T with the metric ρ given by ρ(ξ, η) = e ⁻ⁿ if ξ, η ∈ ∂T intersect in a path of length precisely n from 0.

Math. Institute, the Hebrew University, Jerusalem. Partially sponsored by a grant from the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany)

Math. Dept., Stanford University, Stanford, California 94305. Supported by a Weizmann Postdoctoral fellowship