LBan-III

Goal

The goal of this lecture is to present the final lecture on KPZ and directed polymers in finite dimension. We show that for $d>2$, a "glass transition" occurs.

KPZ: from 1D to the Cayley tree

Much is known about KPZ, but several aspects remain elusive:

In $d=1$, we have $\theta =1/3$ and a glassy regime present at all temperatures. The stationary solution of the KPZ equation describes, at long times, the fluctuations of quantities such as $E_{\min }[x]-E_{\min }[x']$. However, it does not determine the full distribution of $E_{\min }$ for a given $x$. In particular, the origin of the Tracy–Widom distribution remains unclear.

In $d=\infty$, the Cayley tree can be solved exactly, predicting a freezing transition to a 1RSB phase ($\theta =0$).

In finite dimensions greater than one, exact solutions are not available. Numerical simulations indicate $\theta >0$ in $d=2$, while the case $d>2$ remains particularly intriguing.