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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 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 } , so 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 {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 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 a sum of random variables that become heavy tailed 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 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 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 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, 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} }

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

2. Heavy tails and freezing. 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 < T_c } the distribution 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 z } becomes heavy tailed. What does this imply for the sum 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 } ? How fast does it scale with 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 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: 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}. }

   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 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 } , the average IPR equals to:

   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)}. }

   Check this identity numerically (with your favourite program: mathematica, python...). Discuss how this quantity changes across the transition at 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=1 } , and how this fits with what you expect in general in a localized phase.

Problem H.2: Directed polymer on the Bethe Lattice: the replica calculation

Consider the partition function of the directed polymer on the Bethe lattice [......]. For a lattice of length L, the partition function is

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_{\text{P path }\in \mathcal{P}} \prod_{s \in P }e^{-\beta \epsilon_s} }

where 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{P}} is the set of all directed paths on the lattice that go from the root to the leaves at distance L, s are the sites along the path 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 \epsilon_s} the corresponding on-site energy. We assume that these energies are independent, extracted from a distribution 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(\epsilon)} . As usual, we denote with 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{\cdot} } the average with respect to this distribution.

1. Annealed free energy. Compute the annealed free energy of the model for general 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(\epsilon)} .

2. The 1RSB calculation: setting up. We now compute the quenched free energy of the model within the 1RSB ansatz.
• Write the general expression 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 \overline{Z^n}}
• 1RSB ansatz: assume that the n paths we are summing over are organized into m distinct groups 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 n/m } paths; 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 n/m } paths in each group are overlapping from the root of the lattice up to a given length 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 L q_1 } , and then depart up to the end of the lattice (see sketch). Show that the number of distinct configurations of this type is 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 K^{L q_1 m} K^{L (1-q_1) n} } .
• What are the two possible values of overlaps between replicas within this ansatz? What is the probability that two replicas have overlap 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 q_1 } ?
• Show that under the 1RSB assumption it holds: 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{Z^n}= \sum_{q_1, m} K^{L q_1 m}\, \left(\overline{e^{-\beta \frac{n}{m} \epsilon}} \right)^{L q_1 m} \, K^{L (1-q_1) n} \, \left(\overline{e^{-\beta \epsilon}} \right)^{L (1-q_1) n} }

  For which values 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 q_1, m} one would reproduce the annealed calculation?

• Assume 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 m= n/x + O(n^2) } 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 q_1= 1 + O(n) } . Using the replica trick, show that the 1RSB free energy is: 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_{1RSB}=-\frac{1}{\beta} \text{extremum}_x \left[ \frac{1}{x} \, \log \left(K \int d\epsilon \, p(\epsilon) e^{-\beta x \epsilon} \right)\right] } For which values of x this coincide with the annealed free energy?

3. The saddle point. We now compute .

Overlap distribution