KPZ-type fluctuation exponents for interacting diffusions in equilibrium

Abstract

We consider systems of N diffusions in equilibrium interacting through a potential V. We study a “height function,” which, for the special choice $\mathit{V}(\mathit{x})={\mathrm{e}}^{-\mathit{x}}$, coincides with the partition function of a stationary semidiscrete polymer, also known as the (stationary) O’Connell–Yor polymer. For a general class of smooth convex potentials (generalizing the O’Connell–Yor case), we obtain the order of fluctuations of the height function by proving matching upper and lower bounds for the variance of order ${\mathit{N}}^{2/3}$, the expected scaling for models lying in the KPZ universality class. The models we study are not expected to be integrable, and our methods are analytic and nonperturbative, making no use of explicit formulas or any results for the O’Connell–Yor polymer.