Central moments of the free energy of the O'Connell-Yor polymer

TL;DR

This paper derives exact formulas relating cumulants of the free energy of the O'Connell-Yor polymer to boundary jump times, providing near-optimal estimates for moments that confirm KPZ universality class behavior.

Contribution

It develops Gaussian integration by parts formulas to connect free energy cumulants with boundary jump expectations, enabling precise moment estimates.

Findings

Variance growth follows the KPZ exponent 2/3.

Derived formulas relate free energy cumulants to boundary jump moments.

Established near-optimal moment bounds for the free energy and boundary jump times.

Abstract

Sepp\"al\"ainen and Valk\'o showed in \cite{SV} that for a suitable choice of parameters, the variance growth of the free energy of the stationary O'Connell-Yor polymer is governed by the exponent $2/3$, characteristic of models in the KPZ universality class. We develop exact formulas based on Gaussian integration by parts to relate the cumulants of the free energy, $\log Z_{n,t}^\theta$, to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, $s_0$. We then use these formulas to obtain estimates for the $k$-th central moment of $\log Z_{n,t}^\theta$ as well as the $k$-th annealed moment of $s_0$ for $k> 2$, with nearly optimal exponents $(1/3)k+\epsilon$ and $(2/3)k+\epsilon$, respectively.