Uniform asymptotics of Weil-Petersson volumes and their coefficients

This preprint answers two asymptotic questions posed by Mirzakhani in Section 1.4 of her 2013 paper on Weil-Petersson volumes and random hyperbolic surfaces. As the genus tends to infinity, it gives explicit relative asymptotics for the volume ratio V(g,n)/V(g-1,n+2), uniformly over all nonnegative integer cusp counts, and for adjacent coefficients of the one-boundary volume polynomial, uniformly over all valid degrees. An explicit absolute volume model admits an inverse-genus expansion with a controlled remainder through orders proportional to the square root of the genus, uniformly in the cusp count; a sufficiently small square-root truncation has exponentially small relative error in the square root of the genus. The coefficient ratios have a uniform power error bound. Matching with pure psi intersection numbers identifies the universal constant in the Mirzakhani-Zograf large-genus expansion as 1/sqrt(pi), as conjectured by Zograf and previously derived from the nonperturbative topological recursion under a resurgence assumption; the proof here is unconditional. When the cusp count grows faster than the square root of the genus and its ratio to the genus converges in the extended nonnegative real line, applications include Poisson statistics for short primitive closed geodesics and the joint length and cusp-type distribution of the systole. Upper bounds for a growing number of small eigenvalues hold throughout the former growth regime. The full diagonal volume chain is determined on a logarithmic scale.