Eigenvalue Statistics of One-Face Maps

TL;DR

This paper investigates the eigenvalue statistics of adjacency matrices from one-face maps, revealing genus-dependent behaviors and connections to random matrix theory, with implications for understanding map rigidity.

Contribution

It introduces an algorithm for generating genus zero maps and uncovers how eigenvalue statistics vary with genus and map size.

Findings

Eigenvalue statistics of genus zero maps differ from higher genus.

Statistics align with larger classes from Random Matrix Theory.

Distribution of eigenvalue spacings shifts with map size and genus.

Abstract

We examine the adjacency matrices of three-regular graphs representing one-face maps. Numerical studies reveal that the limiting eigenvalue statistics of these matrices are the same as those of much larger, and more widely studied classes from Random Matrix Theory. We present an algorithm for generating matrices corresponding to maps of genus zero, and find the eigenvalue statistics in the genus zero case differ strikingly from those of higher genus. These results lead us to conjecture that the eigenvalue statistics depend on the rigidity of the underlying map, and the distribution of scaled eigenvalue spacings shifts from that of the Gaussian Orthogonal Ensemble to the exponential distribution as the map size increases relative to the genus.