The spectrum of local random Hamiltonians

TL;DR

This paper investigates the spectral properties of local random Hamiltonians, introducing a novel mathematical framework using $$-free convolution and combinatorial structures to analyze eigenvalues.

Contribution

It establishes an isomorphism between $$-noncrossing partitions and permutations, providing new tools to study the spectrum of local random Hamiltonians.

Findings

Derived bounds for the largest eigenvalue of the Hamiltonian

Represented the spectrum using $$-free convolution of local terms

Connected combinatorial structures to spectral analysis

Abstract

The spectrum of a local random Hamiltonian can be represented generically by the so-called $\epsilon$-free convolution of its local terms' probability distributions. We establish an isomorphism between the set of $\epsilon$-noncrossing partitions and permutations to study its spectrum. Moreover, we derive some lower and upper bounds for the largest eigenvalue of the Hamiltonian.