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 do replica!

To make progress in disordered systems we have to go through the moments of the partition function. We recall that

• $V(x,\tau )$ is a Gaussian field with

${\overline {V(x,\tau )}}=0,\quad {\overline {V(x,\tau )V(x',\tau ')}}=D\delta (x-x')\delta (\tau -\tau ')$

• From the Wick theorem, for a generic Gaussian $W$ field we have

\overline{e^W}= e^{\overline{W} +\frac{1}{2} (\overline{W^2}-\overline{W}^2)}

The first moment of the partition function is

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

The term Failed to parse (syntax error): {\displaystyle \overline{W^2} =\int d \tau_1 d\tau_2 \overline{V(x,\tau_1}V(x,\tau_2}= D t \delta(0)} has a short distance divergence due to the Delta functiton, but is path independent. Hence we can write:

${\overline {\exp \left[-{\frac {\int d\tau V(x,\tau )}{T}}\right]}}$

has a divergence at short distances (but on  the lattice model is regularized

and 

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