Anomalous threshold behavior of long range random walks

TL;DR

This paper investigates the threshold behavior of symmetric Markov chains with heavy-tailed jumps on weighted graphs satisfying sub-Gaussian estimates, generalizing classical second moment conditions.

Contribution

It establishes a new threshold phenomenon for heavy-tailed Markov chains when the tail index matches the walk dimension on sub-Gaussian graphs.

Findings

Identifies a critical threshold where jump tail heaviness affects chain behavior

Generalizes classical second moment threshold to sub-Gaussian graph settings

Provides theoretical framework for analyzing heavy-tailed random walks

Abstract

We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk. On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian estimate. In a certain sense, this generalizes the classical threshold corresponding to the second moment condition.