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. The first moment provide the annealed average and the second moment tell us about the fluctuantions. In particular we saw that if

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

then the partition function is self-averaging. In this case annealed and quenched averages are equivalent in the thermodynamic limit. The previous condition is sufficient for the equivalence but not necessary. It is enough to show 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:

Since the disorder is Gaussian, averages of exponentials can be computed using Wick’s theorem:

${\overline {\exp(W)}}=\exp \!{\Big [}{\overline {W}}+{\frac {1}{2}}{\big (}{\overline {W^{2}}}-{\overline {W}}^{2}{\big )}{\Big ]},$

for any Gaussian random variable $W$.

These two properties are all we need to carry out the replica calculation below.