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 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 N \to \infty } . Show that this is the case by computing the second moment 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 \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 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 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 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 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 \overline{\mathcal{N}(E)}} ? Why is the entropy $\Sigma (\epsilon )$, controlling the typical value of 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 \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 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 } 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

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 = \sum_{\alpha=1}^{2^N} e^{-\beta E_\alpha}= \int dE \, \mathcal{N}(E) e^{-\beta E} \equiv e^{-\beta N f + o(N)}. }

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

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 = \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 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 N } , the integral defining $Z$ 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

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 f = -\frac{1}{\beta}\lim_{N \to \infty} \frac{\log Z}{N} = \begin{cases} &- \left( T \log 2 + \frac{1}{4 T}\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

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 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_\alpha } and show that for $(\delta E)^{2}/N\ll 1$ 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,

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 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:

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 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 } is power law distributed with exponent $\mu$, 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 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 p(\omega)= C 2^{-N} (1-\omega)^{\mu-1}\omega^{-1 -\mu} } for $\omega \gg 2^{-{\frac {N}{\mu }}}$, 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