Probability distributions for directed polymers in random media with correlated noise

TL;DR

This paper investigates how spatial correlations in noise affect the probability distribution of free energy in directed polymers, revealing a continuous transition from Tracy-Widom to Gaussian distributions as correlations increase.

Contribution

It demonstrates that correlated noise modifies the universal Tracy-Widom distribution, leading to a new class of interpolating distributions between Tracy-Widom and Gaussian.

Findings

Distribution remains Tracy-Widom for weak correlations

Distribution becomes symmetric and Gaussian-like with strong correlations

Scaling exponent for width matches previous theoretical predictions

Abstract

The probability distribution for the free energy of directed polymers in random media (DPRM) with uncorrelated noise in $d=1+1$ dimensions satisfies the Tracy-Widom distribution. We inquire if and how this universal distribution is modified in the presence of spatially correlated noise. The width of the distribution scales as the DPRM length to an exponent $\beta$, in good (but not full) agreement with previous renormalization group and numerical results. The scaled probability is well described by the Tracy-Widom form for uncorrelated noise, but becomes symmetric with increasing correlation exponent. We thus find a class of distributions that continuously interpolates between Tracy-Widom and Gaussian forms.