Equality of Lifshitz and van Hove exponents on amenable Cayley graphs

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.

References in corpus (3)

• Hamiltonians on discrete structures: Jumps of the integrated density of states and uniform convergence
• Sharpness of the phase transition and exponential decay of the subcritical cluster size for percolation on quasi-transitive graphs
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Cited by in corpus (6)

• A Banach space-valued ergodic theorem and the uniform approximation of the integrated density of states
• Lifshits tails in the hierarchical Anderson model
• Random colourings of aperiodic graphs: Ergodic and spectral properties
• Uniform existence of the integrated density of states for combinatorial and metric graphs over Z^d
• Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs
• Lifshitz asymptotics for percolation Hamiltonians