Uniform large-genus asymptotics of Weil-Petersson volumes
Let $V_{g,n}$ be the Weil--Petersson volume of the moduli space of hyperbolic surfaces of genus $g$ with $n$ cusps. The fixed-$n$ large-genus expansion of Mirzakhani and Zograf has correction terms polynomial in $n$ of degree $2i$ at order $g^{-i}$, and ceases to be uniform when $n$ is of order $\sqrt g$. We prove an asymptotic expansion in powers of $1/g$ that holds uniformly in all $n\ge0$, with coefficients analytic in a density parameter that ranges over a compact interval, including the limits $n/g\to0$ and $n/g\to\infty$. Its leading term is explicit, contains no undetermined constant and has relative error $O(1/g)$; truncation at an order proportional to $\sqrt g$ gives relative error $O(\exp(-a\sqrt g))$ for some $a>0$. For fixed $n$ this confirms Zograf's conjecture that the constant in the Mirzakhani--Zograf asymptotics is $1/\sqrtπ$. We also determine, at every cusp density, the limits of the volume ratios $V_{g,n}/V_{g-1,n+2}$ and $V_{g,n+1}/(2gV_{g,n})$ and, in all degrees, the asymptotics of the ratios of adjacent coefficients of the one-boundary volume polynomial; this answers two questions of Mirzakhani. The proof deforms mixed $κ_1$--$ψ$ intersection numbers into pure $ψ$ intersection numbers along a family whose recursion has nonnegative coefficients, fixes the overall constant through Painlevé I without assuming resurgence, and recovers individual volumes by a local limit theorem in the number of cusps. As an application, the spectral gap of a Weil--Petersson random surface with $n\gg\sqrt g$ cusps tends to zero in probability; this removes the restriction $n/g\to0$ from a theorem of Shen and Wu.