L-4

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

KPZ : from 1d to the Cayley tree

We know a lot about KPZ, but there is still much to understand:

• In $d=1$, we have found $\theta =1/3$ 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 actual distribution of $E_{\min }$ for a given $x$. In particular, we have no clear understanding of the origin of the Tracy-Widom distribution.

• In $d=\infty$, an exact solution exists for the Cayley tree, predicting a freezing transition to a 1RSB phase ($\theta =0$).

• In finite dimensions greater than one, no exact solutions are available. Numerical simulations indicate $\theta >0$ in $d=2$. The case $d>2$ remains particularly intriguing.

Let's do replica!

To make progress in disordered systems, we need to analyze the moments of the partition function. From Valentina's lecture, recall that if

${\frac {\overline {Z(x,t)^{2}}}{({\overline {Z(x,t)}})^{2}}}=1$

then the partition function is self-averaging, and annealed and quenched averages are equivalent.The condition above is sufficient but not necessary. It is enough that

${\frac {\overline {Z(x,t)^{2}}}{({\overline {Z(x,t)}})^{2}}}<{\text{const}}$,

when $t\to \infty$, to ensure the equivalence between annealed and quenched averages.

In the following, we compute this quantity, which corresponds to a two-replica calculation. For simplicity, we consider polymers starting at $0$ and ending at $x$. We recall that: