Revision as of 15:43, 4 January 2024

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 average of the disorder compute the moments of the partition function.

{\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]{\overline {\int _{0}^{t}\int _{0}^{t}d\tau _{1}d\tau _{2}\exp \left[-{\frac {V(x_{1}(tau_{1}))V(x_{2}(tau_{2}))}{T}}\right]}}

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 <center> <math>
-\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] \overline{\exp\left[- \frac{V(x_1(tau)) V(x_2(tau)) }{T}\right]}
+\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] \overline{\int_0^t \int_0^t d \tau_1 d \tau_2  \exp\left[- \frac{V(x_1(tau_1)) V(x_2(tau_2)) }{T}\right]}
 </math></center>
 =Part 2: Structural glasses=