Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

TL;DR

This paper demonstrates that on amenable Cayley graphs, the Lifshitz and van Hove exponents coincide for adjacency Laplacians, revealing a universal low-energy spectral behavior in these structures.

Contribution

It establishes the equality of Lifshitz and van Hove exponents for adjacency Laplacians on amenable Cayley graphs, extending the understanding of spectral asymptotics.

Findings

Lifshitz and van Hove exponents coincide for adjacency Laplacians.

Universal low-energy spectral behavior for combinatorial Laplacians on quasi-transitive graphs.

Results hold for long-range bond percolation models.

Abstract

We study the low energy asymptotics of periodic and random Laplace operators on Cayley graphs of amenable, finitely generated groups. For the periodic operator the asymptotics is characterised by the van Hove exponent or zeroth Novikov-Shubin invariant. The random model we consider is given in terms of an adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph. The asymptotic behaviour of the spectral distribution is exponential, characterised by the Lifshitz exponent. We show that for the adjacency Laplacian the two invariants/exponents coincide. The result holds also for more general symmetric transition operators. For combinatorial Laplacians one has a different universal behaviour of the low energy asymptotics of the spectral distribution function, which can be actually established on quasi-transitive graphs without an amenability assumption. The latter result holds also for long range bond percolation models.