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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 $f$. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z } can be computed with the saddle point method; show that a transition occurs at a critical temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_\alpha= - N \sqrt{\log 2} + \delta E_\alpha } , and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\delta E)^2/N \ll 1 } this is an exponential. Using this, compute the distribution of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_\alpha } is distributed as $p(\omega )=C2^{-N}(1-\omega )^{\mu -1}\omega ^{-1-\mu }$ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega \gg 2^{-\frac{N}{\mu}} } , and that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle IPR= \frac{\Gamma(2-\mu)}{\Gamma(\mu) \Gamma(1-\mu)} }

This last point: make an homework