Free-energy distribution of the directed polymer at high temperature

Understand

We study the directed polymer of length $t$ in a random potential with fixed endpoints in dimension 1+1 in the continuum and on the square lattice, by analytical and numerical methods.

• The universal regime of high temperature $T$ is described, upon scaling 'time' $t \sim T^5/\kappa$ and space $x = T^3/\kappa$ (with $\kappa=T$ for the discrete model) by a continuum model with $\delta$-function disorder correlation.
• Using the Bethe Ansatz solution for the attractive boson problem, we obtain all positive integer moments of the partition function.
• The lowest cumulants of the free energy are predicted at small time and found in agreement with numerics.