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

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.