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

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

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

$\overline{e^W}=e^{\bar{W}+\frac{1}{2}(\overline{W^2}-\stackrel{2}{\stackrel{‾}{W}})}$

The first moment of the partition function is

$\overline{Z_t[x_1]}=\int {𝒟}{x}_1\exp [-\frac{1}{T}\underset{0}{\overset{t}{\int }}d\tau ({\partial }_{\tau }x{)}^2]\overline{\exp [-\frac{1}{T}\int d\tau V(x,\tau )]}$

Note that 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:

$ciao$ $\overline{Z_t[x_1]Z_t[x_2]}=\int {𝒟}{x}_1\int {𝒟}{x}_2\exp [-\frac{1}{2T}\underset{0}{\overset{t}{\int }}d\tau [({\partial }_{\tau }{x}_1{)}^2+({\partial }_{\tau }{x}_2{)}^2]\underset{0}{\overset{t}{\int }}\underset{0}{\overset{t}{\int }}d{\tau }_{1}d{\tau }_2\overline{\exp [-\frac{V(x_1({\tau }_1))V(x_2({\tau }_2))}{T}]}$