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.

Directed Polymer in finite dimension === State of the Art =

The directed polymer in random media belongs to the KPZ universality class. The behavior of this system is well understood in one dimension and in the mean-field case, more precisely for the directed polymer on the Cayley tree. In particular:

• In $d=1$, we have $\theta =1/3$ and a glassy regime present at all temperatures. The model is integrable through a non-standard Bethe Ansatz, and the distribution of $E_{\min }$ for a given boundary condition is of the Tracy–Widom type.

• In $d=\mathrm{\infty }$, for the Cayley tree, an exact solution exists, 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 $d=2$ and a glassy regime present at all temperatures. 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. To ensure the equivalence between the two averages, it is enough that for large t

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

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: