T-I

Problem 1: the energy landscape of the REM

In this exercise we characterize the energy landscape of the REM, by determining the number ${\mathcal {N}}(E)dE$ of configurations having energy $E_{\alpha }\in [E,E+dE]$. This quantity is a random variable. For large $N$, we will show that its typical value is given by

${\mathcal {N}}(E)=e^{N\Sigma \left({\frac {E}{N}}\right)+o(N)},\quad \quad \Sigma (\epsilon )={\begin{cases}\log 2-\epsilon ^{2}\quad &{\text{ if }}|\epsilon |\leq {\sqrt {\log 2}}\\0\quad &{\text{ if }}|\epsilon |>{\sqrt {\log 2}}\end{cases}}$

The function $\Sigma (\epsilon )$ is the entropy of the model, and it is sketched in Fig. X. The point where the entropy vanishes, $\epsilon =-{\sqrt {\log 2}}$, is the energy density of the ground state, consistently with what we obtained with extreme values statistics. The entropy is maximal at $\epsilon =0$: the highest number of configurations have vanishing energy density.

1. The annealed entropy. We begin by computing the annealed entropy $\Sigma ^{A}$, which is the function that controls the behaviour of the average number of configurations at a given energy, ${\overline {{\mathcal {N}}(E)}}={\text{exp}}\left(N\Sigma ^{A}\left({\frac {E}{N}}\right)+o(N)\right)$. Compute this function using the representation ${\mathcal {N}}(E)dE=\sum _{\alpha =1}^{2^{N}}\chi _{\alpha }(E)dE\;$ [with $\chi _{\alpha }(E)=1$ if $E_{\alpha }\in [E,E+dE]$ and $\chi _{\alpha }(E)=0$ otherwise], together with the distribution $p(E)$ of the energies of the REM configurations. When does $\Sigma ^{A}$ coincide with the entropy defined above?

2. Self-averaging quantities. For $|\epsilon |\leq {\sqrt {\log 2}}$ the quantity ${\mathcal {N}}(E)$ is self-averaging: its distribution concentrates around the average value ${\overline {\mathcal {N}}}(E)$ when $N\to \infty$. Show that this is the case by computing the second moment ${\overline {{\mathcal {N}}^{2}}}$ and using the central limit theorem. Show that this is no longer true in the region where the annealed entropy is negative: why does one expect fluctuations to be relevant in this region?

3. Average vs typical. For $|\epsilon |>{\sqrt {\log 2}}$ the annealed entropy is negative, meaning that the average number of configurations with those energy densities is exponentially small in $N$. This implies that configurations with those energy are exponentially rare: do you have an idea of how to show this, using the expression for ${\overline {{\mathcal {N}}(E)}}$? Why is the entropy $\Sigma (\epsilon )$, controlling the typical value of ${\mathcal {N}}(E)$, zero in this region? Why the point where the entropy vanishes coincides with the ground state energy of the model?

Comment: this analysis of the landscape suggests that fluctuations due to the randomness become relevant when one looks at the bottom of their energy landscape, close to the ground state energy density. We show below that this intuition is correct, and corresponds to the fact that the partition function $Z$ is not self-averaging in the low-T phase of the model.

Problem 2: the free energy and the freezing transition

We now compute the equilibrium phase diagram of the model, and in particular the free energy density $f$, showing that:

$f=-{\frac {1}{\beta }}\lim _{N\to \infty }{\frac {\log Z}{N}}={\begin{cases}&-\left(T\log 2+{\frac {1}{4T}}\right)\quad {\text{if}}\quad T\geq T_{c}\\&-{\sqrt {\log 2}}\quad {\text{if}}\quad T<T_{c}\end{cases}}\quad \quad T_{c}=(2{\sqrt {\log 2}})^{-1}$

At $T_{c}$ a transition occurs, often called freezing transition: in the whole low-temperature phase, the free energy freezes at the value that it has at the critical temperature $T_{c}$.

1. The critical temperature, and the freezing. The partition function the REM reads $Z=\sum _{\alpha =1}^{2^{N}}e^{-\beta E_{\alpha }}=\int dE\,{\mathcal {N}}(E)e^{-\beta E}.$ Using the behaviour of the typical value of ${\mathcal {N}}$ determined in Problem 1, derive the free energy of the model (hint: perform a saddle point calculation). What is the order of this thermodynamic transition?

2. Entropy, fluctuations, annealed vs quenched. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? Similarly to the entropy, one can define an annealed free energy $f^{A}$ from ${\overline {Z}}=e^{-N\beta f^{A}+o(N)}$: when does this coincide with the free energy computed above?

2. Entropy, fluctuations, annealed vs quenched. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? Similarly to the entropy, one can define an annealed free energy $f^{A}$ from ${\overline {Z}}=e^{-N\beta f^{A}+o(N)}$: when does this coincide with the free energy computed above?

Problem 3: freezing, heavy tails, condensation

The freezing transition can also be understood in terms of extreme valued statistics, as discussed in the lecture. Define $E_{\alpha }=-N{\sqrt {\log 2}}+\delta E_{\alpha }$, and

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

• Heavy tails. 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,

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

What happens when $T\to T_{c}$? How does the behaviour of the partition function change at the transition point? Is this consistent with the behaviour of the entropy?

• Inverse participation ratio. The low temperature behaviour of the partition function an be characterized in terms of a standard measure of condensation (or localization), the Inverse Participation Ratio (IPR) defined as:

$IPR={\frac {\sum _{\alpha =1}^{2^{N}}z_{\alpha }^{2}}{[\sum _{\alpha =1}^{2^{N}}z_{\alpha }]^{2}}}=\sum _{\alpha =1}^{2^{N}}\omega _{\alpha }^{2}\quad \quad \omega _{\alpha }={\frac {z_{\alpha }}{\sum _{\alpha =1}^{2^{N}}z_{\alpha }}}$

Show that when $z$ is power law distributed with exponent $\mu$, $\omega _{\alpha }$ is distributed as $p(\omega )=C2^{-N}(1-\omega )^{\mu -1}\omega ^{-1-\mu }$ for $\omega \gg 2^{-{\frac {N}{\mu }}}$, and that

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

This last point: make an homework