Homework

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

In this problem, we show how the freezing transition of the Random Energy Model can be understood in terms of extreme valued statistics 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 (or condensation).

1. Power laws. 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, $p(z)={\frac {c}{z^{1+\mu }}}\quad \quad \mu ={\frac {2{\sqrt {\log 2}}}{\beta }}$

   For which values of temperature the second moment of z exists? And the first moment?

2. Heavy tails and freezing. When $T<T_{c}$ the distribution of $z$ becomes heavy tailed. What does this imply for the sum $Z$? How fast does it scale with $M=2^{N}$? Discuss in which sense this is consistent with the behaviour of the partition function and of the entropy discussed in Problem 1.2. In particular, intuitively, why can one talk about a localization or condensation transition?

3. 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 $\mu$, the average IPR equals to:

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

   Check this identity numerically (with your favourite program: mathematica, python...). 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.

Problem H.2: Directed polymer on the Bethe Lattice with replicas

Consider the partition function of the