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A Correlation Inequality for Tree-Indexed Markov Chains

  • Itai Benjamini
  • Yuval Peres
Part of the Progress in Probability book series (PRPR, volume 29)

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.

Keywords

Simple Random Walk Consecutive Integer Correlation Inequality Rich Tree Burst Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Itai Benjamini
    • 1
  • Yuval Peres
    • 2
  1. 1.Institute of MathematicsHebrew University Givat RamJerusalemIsrael
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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