Random Hyperbolic Surfaces Achieve Optimal Spectral Gap 1/4 — E8 Intelligence Research

FINDING: Random compact hyperbolic surfaces exhibit an optimal spectral gap λ₁ → 1/4 (Selberg bound) as genus → ∞, with Benjamini-Schramm convergence as the limiting notion of "local convergence" of the underlying geometry. | MATH: For a hyperbolic surface X = Γ\ℍ², the Laplacian spectrum on L²(X) has bottom λ₀ = 0; the spectral gap is λ₁. Selberg's 3/16 theorem gives λ₁ ≥ 3/16 for arithmetic congruence covers; the conjectured optimal bound is λ₁ ≥ 1/4 (the continuous-spectrum threshold of ℍ²). Monk–Thomas (and earlier Mirzakhani, Petri) show that for the Weil–Petersson random model of genus-g surfaces, λ₁ → 1/4 in probability as g → ∞. Benjamini-Schramm convergence: a sequence of finite-area hyperbolic surfaces Xₙ → (ℍ², o) in the BS sense if for every R > 0, Vol({x ∈ Xₙ : inj(x) < R}) / Vol(Xₙ) → 0. Plancherel convergence is strictly stronger than BS convergence (arXiv:2407.18822), characterized by convergence of length-spectrum counting functions. | CONNECTION: The critical constant

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com