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.

Directed polymers from 1D to the Cayley tree

• In $d=1$, we have $\theta =1/3$ and a glassy regime present at all temperatures. The full distribution of $E_{\min }$ for a given $x$ is in the Tracy–Widom family.

• In $d=\mathrm{\infty }$, 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.

Replica Analysis

To study disordered systems, we analyze moments of the partition function. From the first lecture, recall that if