Airy limit for the Jack process and topological expansion

Abstract

For every $\beta>0$, we establish multi-time soft-edge moment convergence for the Jack--Plancherel process, a discrete $\beta$-analogue of Dyson Brownian motion. The limiting moments admit an absolutely convergent expansion in terms of nonnegative Brownian bridges decorated by pairs of equal and opposite jumps. At equal times, we identify this limit with the joint Laplace statistics of the Airy$_\beta$ process for $\beta\geq1$. The same Brownian expansion yields an asymptotic $\beta$-topological expansion of the marginal $b$-conjecture and, at $\beta=2$, an explicit nonnegative formula for the Witten--Kontsevich intersection numbers.