Trapping in the random conductance model

TL;DR

This paper studies how random walks in a random conductance environment tend to get trapped in small regions when returning to the start, especially in higher dimensions where subdiffusive behavior occurs.

Contribution

It demonstrates that trapping dominates the behavior of the walk in subdiffusive regimes and confirms a conjecture about the slowest decay in four dimensions.

Findings

Walks get trapped for order n time in small regions in subdiffusive cases

Trapping explains the subdiffusive decay of the heat kernel

Confirms the worst-case subdiffusive decay in four dimensions

Abstract

We consider random walks on $\Z^d$ among nearest-neighbor random conductances which are i.i.d., positive, bounded uniformly from above but whose support extends all the way to zero. Our focus is on the detailed properties of the paths of the random walk conditioned to return back to the starting point at time $2n$. We show that in the situations when the heat kernel exhibits subdiffusive decay --- which is known to occur in dimensions $d\ge4$ --- the walk gets trapped for a time of order $n$ in a small spatial region. This shows that the strategy used earlier to infer subdiffusive lower bounds on the heat kernel in specific examples is in fact dominant. In addition, we settle a conjecture concerning the worst possible subdiffusive decay in four dimensions.