Homework

Problem H.1: freezing as a localization/condensation transition

In this final exercise, we show how the freezing transition can be understood in terms of extreme valued statistics (discussed in the lecture) and localization. We consider the energies of the configurations and define $E_{\alpha }=-N{\sqrt {\log 2}}+\delta E_{\alpha }$, so that

${Z}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}e^{-\beta \delta E_{\alpha }}=e^{\beta N{\sqrt {\log 2}}}\sum _{\alpha =1}^{2^{N}}z_{\alpha }$

We show that $Z$ is a sum of random variables that become heavy tailed for $T<T_{c}$, implying that the central limit theorem is violated and this sum is dominated by few terms, the largest ones. This can be interpreted as the occurrence of localization.

1. Heavy tails and concentration. Compute the distribution of the variables $\delta E_{\alpha }$ and show that for $(\delta E)^{2}/N\ll 1$ this is an exponential. Using this, compute the distribution of the $z_{\alpha }$ and show that it is a power law, $p(z)={\frac {c}{z^{1+\mu }}}\quad \quad \mu ={\frac {2{\sqrt {\log 2}}}{\beta }}$ When $T<T_{c}$, one has $\mu <1$: the distribution of $z$ becomes heavy tailed. What does this imply for the sum $Z$? Is this consistent with the behaviour of the partition function and of the entropy discussed in Problem 2? Why can one talk about a localization or condensation transition?

2. Inverse participation ratio. The low temperature behaviour of the partition function an be characterized in terms of a standard measure of localization (or condensation), the Inverse Participation Ratio (IPR) defined as: $IPR={\frac {\sum _{\alpha =1}^{2^{N}}z_{\alpha }^{2}}{[\sum _{\alpha =1}^{2^{N}}z_{\alpha }]^{2}}}=\sum _{\alpha =1}^{2^{N}}\omega _{\alpha }^{2}\quad \quad \omega _{\alpha }={\frac {z_{\alpha }}{\sum _{\alpha =1}^{2^{N}}z_{\alpha }}}.$

   When $z$ is power law distributed with exponent $\mu$, one can show (HOMEWORK!) that

   $IPR={\frac {\Gamma (2-\mu )}{\Gamma (\mu )\Gamma (1-\mu )}}.$

   Discuss how this quantity changes across the transition at $\mu =1$, and how this fits with what you expect in general in a localized phase.