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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 <math> E_\alpha= - N \sqrt{\log 2} + \delta E_\alpha </math>, so that
<center><math>
{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
</math></center>
We show that <math> Z </math> is a sum of random variables that become heavy tailed for <math> T < T_c </math>, 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).
<ol>
<li> <em> Power laws.</em> Compute the distribution of the variables <math> \delta E_\alpha </math> and show that for <math> (\delta E)^2/N \ll 1 </math> this is an exponential. Using this, compute the distribution of the  <math> z_\alpha </math> and show that it is a power law,
<center><math>
p(z)= \frac{c}{z^{1+\mu}} \quad \quad \mu= \frac{2 \sqrt{\log 2}}{\beta}
</math></center>
For which values of temperature the second moment of z exists? And the first moment?
</li>
</ol>
<br>
<ol start="2">
<li> <em> Heavy tails and freezing. </em>When <math> T < T_c </math> the distribution of  <math> z </math> becomes heavy tailed. What does this imply for the sum <math> Z </math>? How fast does it scale with <math> M=2^N </math>? 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?
</li>
</ol>
<br>
<ol start="3">
<li><em> Inverse participation ratio.</em> 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:
<center><math>
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}.
</math></center>
When <math> z </math> is power law distributed with exponent <math> \mu </math>, the average IPR equals to:
<center><math>
IPR= \frac{\Gamma(2-\mu)}{\Gamma(\mu) \Gamma(1-\mu)}.
</math></center>
Check this identity numerically (with your favourite program: mathematica, python...). Discuss how this quantity changes across the transition at <math> \mu=1 </math>, and how this fits with what you expect in general in a localized phase. 
</li>
</ol>
<br>
=== Problem H.2: Directed polymer on the Bethe Lattice: the replica calculation ===
[[File:Bethe.png|thumb|left|x120px|Bethe lattice of depth L=4 and branching K=2. A random energy <math> \epsilon </math> is associated to each site of the lattice.]]
Consider the partition function of the directed polymer on the Bethe lattice [......]. For a lattice of depth L, the partition function is
<center><math>
Z= \sum_{\text{P path }\in \mathcal{P}} \prod_{s \in P }e^{-\beta \epsilon_s}
</math></center>
where <math>\mathcal{P}</math> 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 <math>\epsilon_s</math> the corresponding on-site energy. We assume that these energies are independent, extracted from a distribution  <math>p(\epsilon)</math>. As usual, we denote with <math> \overline{\cdot} </math> the average with respect to this distribution.
<ol>
<li> <em> Annealed free energy.</em> Compute the annealed free energy of the model for general  <math>p(\epsilon)</math>.
</li>
</ol>
<br>
[[File:Bethe 1RSB.png|thumb|right|x142px|Sketch of 1RSB ansatz: the <math> n=9 </math> paths are organized into <math> m=3 </math> groups of <math> n/m=3 </math> paths each. In each group, the paths overlap for a length <math> Lq_1= 3 </math> (blue parts of the paths) and then depart for the remaining length (orange part of the paths). ]]
<ol start="2">
<li> <em> The quenched free energy: 1RSB. </em> We now compute the quenched free energy of the model within the 1RSB ansatz. </li>
<ul>
<li>  Write the general expression of <math> \overline{Z^n}</math>  </li>
<li>  1RSB ansatz: assume that the <math>n</math> paths we are summing over are organized into <math> n/m </math> distinct groups of <math> m </math> paths; the <math> m </math> paths in each group are overlapping from the root of the lattice up to a given length <math> L q_1 </math>, and then depart up to the end of the lattice (see sketch). Show that the number of distinct configurations of this type is <math> K^{L q_1 \frac{n}{m}} K^{L (1-q_1) n} </math>.</li>
<li>  What are the two possible values of overlaps between replicas within this ansatz? What is the probability that two replicas have overlap <math> q_1 </math>?</li>
<li> Show that under the 1RSB assumption it holds:
<center><math>
\overline{Z^n}= \sum_{q_1, m} K^{L q_1 \frac{n}{m}}\,  \left(\overline{e^{-\beta m \epsilon}} \right)^{L q_1 \frac{n}{m}} \,  K^{L (1-q_1) n} \,  \left(\overline{e^{-\beta \epsilon}} \right)^{L (1-q_1) n}
</math></center>
</li>
<li> Assume that <math> q_1= 1 - n \delta q_1 </math>. Using the replica trick, show that the 1RSB free energy is:
<center><math>
f_{1RSB}=-\frac{1}{\beta} \text{min}_{m \in [0,1]} \left[ \frac{1}{m} \, \log \left(K \int d\epsilon \, p(\epsilon) e^{-\beta m \epsilon} \right)\right]
</math></center>
Why is <math> m </math>  restricted to that range? For which values of <math> m </math>  this coincide with the annealed free energy? </li>
</ul>
</ol>
<br>
<ol start="3">
<li> <em> The transition. </em> Consider now a Gaussian distribution of on-site energies, <math> p(\epsilon)=e^{-\epsilon^2/2}/\sqrt{2 \pi}</math>. </li>
<ul>
<li>Compute the value <math>m^*</math> which satisfies
<center><math>
\partial_m \left[ \frac{1}{m} \, \log \left(K \int d\epsilon \, p(\epsilon) e^{-\beta m \epsilon} \right)\right]\Big|_{m^*}=0
</math></center>
and show that there is a critical temperature <math> T_c = 1/\sqrt{2 \log K}</math> above which <math>m^*</math> is not in the range <math>[0,1]</math>: which solution should one take for <math> T>T_c</math>?
Express <math> m^* </math> in terms of <math> T_c</math>.  </li>
<li>Compute the free energy above and below the transition. When does the quenched free energy coincide with the annealed?</li>
<li>Why is this transition analogous to the freezing transition of the Random Energy model discussed in Problems 1? </li>
<li>Compute the overlap distribution and explain in which sense the parameter <math>m </math> can be interpreted as a probability. How is the overlap distribution? </li>
</ul>
</ol>
<br>
<ol start="4">
<li> <em> Simulations?. </em> Esercizio numerico di Vicio? </li>
<ul>
</ul>
</ol>
<br>
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Latest revision as of 12:22, 4 February 2024

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