Tracy–Widom fluctuations of the spectral gap for random hyperbolic surfaces

Establish that, for suitable models of random hyperbolic surfaces, the normalized fluctuation ((1/4) − λ1(X))·Vol(X)^{2/3}, after re-scaling by a constant, converges in distribution to the Tracy–Widom law with β = 1.

Background

Motivated by analogies with random matrix theory (GOE) and recent breakthroughs on random regular graphs showing Tracy–Widom fluctuations of the spectral edge, the authors expect similar behavior for Laplacian spectral gaps on random hyperbolic surfaces.

They suggest that proving such a convergence would not only pinpoint the fluctuation scale but also imply a positive proportion of surfaces are “Selberg” (i.e., with spectral gap at least 1/4), thereby addressing the main open problem about Selberg surfaces.