High temperature behaviors of the directed polymer on a cylinder
Abstract
In this paper, we study the free energy of directed polymers on a cylinder of radius $L$ with the inverse temperature $\beta$. Assuming the random environment is given by a Gaussian process that is white in time and smooth in space, with an arbitrary compactly supported spatial covariance function, we obtain precise scaling behaviors of the limiting free energy for high temperatures $\beta\ll1$, followed by large $L\gg1$, in all dimensions. Our approach is based on a perturbative expansion of the PDE hierarchy satisfied by the multipoint correlation function of the polymer endpoint distribution. We also study the case where the random environment is given by the $1+1$ spacetime white noise, and derive an explicit expression of the limiting free energy.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07368
 Bibcode:
 2021arXiv211007368G
 Keywords:

 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 17 pages