Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\geq 3$

Abstract

We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 Δu_{\eps,t} d t+ β\eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\geq 3$, 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 $\barβ\in (0,\infty)$ so that the solution exhibits weak disorder when $β<\barβ$ and strong disorder when $β> \barβ$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.