Nearly optimal spectral gaps for random Belyi surfaces

TL;DR

This paper proves that random hyperbolic surfaces in the Brooks-Makover model have spectral gaps close to the optimal bound, confirming a long-standing conjecture in the field.

Contribution

It establishes a nearly optimal lower bound for the spectral gap of random Belyi surfaces, advancing understanding of their spectral properties.

Findings

Spectral gap exceeds rac{1}{4} minus a small term

Confirms the nearly optimal spectral gap conjecture

Provides bounds for spectral gaps in the Brooks-Makover model

Abstract

In this paper, we show that a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\left(\frac{1}{4}-\left(\frac{1}{n}\right)^{\frac{1}{221}}\right)$, confirming the nearly optimal spectral gap conjecture in this model.