On the spectral gap of the path graph in the limit of large volume

TL;DR

This paper investigates how the spectral gap of the path graph behaves in the large-volume limit under an external potential, revealing it closes faster than in the free case due to increased volume and ground state degeneracy.

Contribution

It demonstrates the asymptotic behavior of the spectral gap of the path graph with an external potential, extending continuous setting results to discrete graphs.

Findings

Spectral gap converges to zero faster than for the free Laplacian.

Large volume and potential cause effective ground state degeneracy.

The mechanism involves combined effects of volume increase and degeneracy.

Abstract

In this paper we study the spectral gap of the path graph and illustrate an interesting effect which has been described recently in the continuous setting. More explicitly, in the large-volume limit and in the presence of a certain external potential, it is shown that the spectral gap converges to zero strictly faster than it does for the free Laplacian. The underlying mechanism is a combination of the increase in volume and an effective degeneracy of the ground state in the limiting regime.