Central moments of the free energy of the stationary O’Connell–Yor polymer

Abstract

Seppäläinen and Valkó showed in (ALEA Lat. Am. J. Probab. Math. Stat. 7 (2010) 451–476) 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, $log{\mathit{Z}}_{\mathit{n},\mathit{t}}^{\mathit{\theta }}$, to expectations of products of quenched cumulants of the time of the first jump from the boundary into the system, ${\mathit{s}}_0$. We then use these formulas to obtain estimates for the kth central moment of $log{\mathit{Z}}_{\mathit{n},\mathit{t}}^{\mathit{\theta }}$ as well as the kth annealed moment of ${\mathit{s}}_0$ for $\mathit{k}>2$, with nearly optimal exponents $(1/3)\mathit{k}+\mathit{ϵ}$ and $(2/3)\mathit{k}+\mathit{ϵ}$, respectively.

As an application, we derive new high probability bounds for the distance between the polymer path and a straight line connecting the origin to the endpoint of the path.