Convergences of the rescaled Whittaker stochastic differential equations and independent sums

Abstract

We study some SDEs derived from the $\mathit{q}\to 1$ limit of a 2D surface growth model called the q-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that “come down from infinity”: After rescaling and re-centering, convergences to the time-inverted stationary additive stochastic heat equation (SHE) hold. The point of view in this paper is a novel probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations, both in diverging integrated forms, explain the convergence of the re-centered covariance functions. The proof of the process-level convergence identifies additional divergent terms in the dynamics and considers nontrivial cancellations.