Nearly optimal spectral gaps for random Belyi surfaces

Abstract

In this paper, we show that a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\left(\frac{1}{4}-\frac{c}{\log n}\right)$ for some universal constant $c>0$ , confirming the nearly optimal spectral gap conjecture in this model.

50 pages, 9 figures, add an Appendix