Aging and sub-aging for one-dimensional random walks amongst random conductances

TL;DR

This paper investigates the long-term behavior of one-dimensional random walks in random conductances with heavy-tailed distributions at zero and/or infinity, revealing aging phenomena and their dependence on conductance tail properties.

Contribution

It provides new aging results for random walks with conductances having heavy tails at zero and/or infinity, including classical aging and sub-aging behaviors.

Findings

Aging observed for the maximum of the process when conductances are heavy-tailed at zero.

Classical aging for the walk's position when heavy tails are at infinity.

Sub-aging phenomena occurring on shorter time-scales with heavy tails at both zero and infinity.

Abstract

We consider random walks amongst random conductances in the cases where the conductances can be arbitrarily small, with a heavy-tailed distribution at 0, and where the conductances may or may not have a heavy-tailed distribution at infinity. We study the long time behaviour of these processes and prove aging statements. When the heavy tail is only at 0, we prove that aging can be observed for the maximum of the process, i.e. the same maximal value is attained repeatedly over long time-scales. When there are also heavy tails at infinity, we prove a classical aging result for the position of the walker, as well as a sub-aging result that occurs on a shorter time-scale.