Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in d ≥ 3

Abstract

We consider the smoothed multiplicative noise stochastic heat equation d u_{\eps,t}= \frac 12 \Delta u_{\eps,t} d t+ \beta \eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1, in dimension d≥3, where $B_{\eps,t}$ is a spatially smoothed (at scale $\eps$) space-time white noise, and β>0 is a parameter. We show the existence of a β¯∈(0,∞) so that the solution exhibits weak disorder when β<β¯ and strong disorder when β>β¯. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.