Spectral dimension of simple random walk on a long-range percolation cluster

TL;DR

This paper determines the spectral dimension of simple random walk on long-range percolation clusters on rom the model's parameters, providing new heat kernel bounds and extending results beyond nearest-neighbor cases.

Contribution

It establishes the spectral dimension for long-range percolation clusters in the regime s>d, introduces new heat kernel bounds, and extends results to non-nearest-neighbor models.

Findings

Spectral dimension explicitly determined for s>d, except at a critical point.

New on-diagonal heat kernel bounds derived, including lower bounds.

Extension of heat kernel bounds to the range s,2d, beyond nearest-neighbor models.

Abstract

Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e.\ for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters $s\in (d,2d)$. We further note that our approach is applicable to short-range models as well.