Uniform large-genus asymptotics of Weil-Petersson volumes

Source abstract

Let $V_{g,n}$ denote the Weil--Petersson volume of the moduli space of genus-$g$ hyperbolic surfaces with $n$ cusps. We obtain an asymptotic expansion in inverse powers of the genus as $g\to\infty$, uniformly for all $n\ge0$. This includes cusp counts of order $\sqrt g$ and larger, where the fixed-$n$ expansion of Mirzakhani and Zograf no longer gives a uniform approximation. The leading term is explicit and has relative error $O(g^{-1})$. Truncating the expansion at a suitably chosen order proportional to $\sqrt g$ gives relative error $O(\exp(-a\sqrt g))$ for some $a>0$, with no restriction on the growth of $n$. For fixed $n$, the leading term identifies the universal constant in the Mirzakhani--Zograf asymptotics as $1/\sqrtπ$, as conjectured by Zograf. We also answer two questions of Mirzakhani: we determine explicit asymptotic profiles for volume ratios uniformly in the number of cusps, and for adjacent coefficient ratios of the one-boundary volume polynomial uniformly over all degrees. The proof uses a positive recursion for mixed $κ_1$--$ψ$ intersection numbers, with the absolute normalization determined by comparison with the known asymptotics of pure $ψ$ intersection numbers.