L-4

Goal 1: final lecture on KPZ and directed polymers at finite dimension. We will show that for $d>2$ a "glass transition" takes place.

Goal 2: We will mention some ideas related to glass transition in true glasses.

Part 1: KPZ in finite dimension

• In $d=1$ we found $\theta =1/3$ and a glassy regime present at all temperatures. Moreover, the stationary solution tell us that $E_{\min }[x]$ is a Brownian motion in $x$. However this solution does not identify the actual distribution of $E_{\min }$ for a given $x$. In particular we have no idea from where Tracy Widom comes from.

• In $d>1$ the exponents are not known. There is an exact solution for the Caley tree (infinite dimension) that predicts a freezing transition to an 1RSB phase ($\theta =0$).

Let's average of the disorder compute the moments of the partition function.

${\overline {Z_{t}[x_{1}]Z_{t}[x_{2}]}}=\int {\cal {D}}x_{1}\int {\cal {D}}x_{2}\exp \left[-{\frac {1}{2T}}\int _{0}^{t}d\tau [(\partial _{\tau }x_{1})^{2}+(\partial _{\tau }x_{2})^{2}\right]\int _{0}^{t}\int _{0}^{t}d\tau _{1}d\tau _{2}{\overline {\exp \left[-{\frac {V(x_{1}(\tau _{1}))V(x_{2}(\tau _{2}))}{T}}\right]}}$