T-I

The REM: the energy landscape

To characterize the energy landscape of the REM, we determine the number ${\mathcal {N}}(E)dE$ of configurations having energy $E_{\alpha }\in [E,E+dE]$. This quantity is a random variable. For large $N$, 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.

• The annealed entropy. The annealed entropy $\Sigma ^{A}$ is a function that controls the behaviour of the average number of configurations at a given energy, ${\overline {{\mathcal {N}}(E)}}=e^{N\Sigma ^{A}\left({\frac {E}{N}}\right)+o(N)}$. To compute it, write ${\mathcal {N}}(E)dE=\sum _{\alpha =1}^{2^{N}}\chi _{\alpha }(E)dE$ with $\chi _{\alpha }(E)=1$ if $E_{\alpha }\in [E,E+dE]$ and $\chi _{\alpha }(E)=0$ otherwise. Use this together with $p(E)$ to obtain $\Sigma ^{A}$ : when does this function coincide with the entropy defined above?

• Self-averaging quantities. For $|\epsilon |\leq {\sqrt {\log 2}}$ the quantity ${\mathcal {N}}(E)$ is self-averaging. This means that 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.

• Average vs typical number. 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?

this will be responsible of the fact that the partition function $Z$ is not self-averaging in the low-T phase, as we discuss below.

The REM: the free energy and the freezing transition

We now compute the equilibrium phase diagram of the model, and in particular the free energy density $f$. The partition function reads

$Z=\sum _{\alpha =1}^{2^{N}}e^{-\beta E_{\alpha }}=\int dE\,{\mathcal {N}}(E)e^{-\beta E}\equiv e^{-\beta Nf+o(N)}.$

We have determined above the behaviour of the typical value of ${\mathcal {N}}$ for large $N$. The typical value of the partition function is therefore

$Z=\int dE\,{\mathcal {N}}(E)e^{-\beta E}=\int d\epsilon \,e^{N\left[\Sigma (\epsilon )-\beta \epsilon \right]+o(N)}.$

• The critical temperature. In the limit of large $N$, the integral defining $Z$ can be computed with the saddle point method; show that a transition occurs at a critical temperature $T_{c}=(2{\sqrt {\log 2}})^{-1}$, and that the free energy density reads

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

• Freezing: the entropy. The thermodynamic transition of the REM is often called a freezing transition. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase?

• Quenched vs annealed free energy. Domination by rare events

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 $T\to T_{c}$?