Asymptotic behaviors of the integrated density of states for random Schr\"odinger operators associated with Gibbs Point Processes

TL;DR

This paper investigates the asymptotic behavior of the integrated density of states for Schrödinger operators linked to Gibbs point processes, revealing similarities and differences with Poisson processes in the low-energy limit.

Contribution

It characterizes the leading terms of the integrated density of states for Gibbs point processes, extending understanding beyond the Poisson case, especially for pairwise interaction models.

Findings

Leading terms match Poisson processes for some Gibbs processes.

Different asymptotics are found for pairwise interaction Gibbs processes.

Provides explicit asymptotic formulas for certain Gibbs point processes.

Abstract

The asymptotic behaviors of the integrated density of states $N(\lambda)$ of Schr\"odinger operators with nonpositive potentials associated with Gibbs point processes are studied. It is shown that for some Gibbs point processes, the leading terms of $N(\lambda)$ as $\lambda\downarrow-\infty$ coincide with that for a Poisson point process, which is known. Moreover, for some Gibbs point processes corresponding to pairwise interactions, the leading terms of $N(\lambda)$ as $\lambda\downarrow-\infty$ are determined, which are different from that for a Poisson point process.