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 $d=\infty$, 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: