T-I

Goal: derive the equilibrium phase diagram of the simplest spin-glass model, the Random Energy Model (REM).
Techniques: saddle point approximation, Legendre transform, probability theory.

A dictionary for large-N disordered systems

• Exponentially scaling variables. We will consider positive random 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 X_N } which depend on a parameter 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 \gg 1 } (the number of degrees of freedom: for a system of size 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} in dimension 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 d} , 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 = L^d } ) and which have the scaling 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 X_N \sim e^{N Y_N} } : this means that the rescaled variable 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 Y_N=N^{-1}\log X_N } has a well defined distribution that remains 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 O(1) } when $N\to \infty$. The standard example we have in mind are the partition functions of disordered systems 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 N } degrees of freedom, 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_N \sim e^{-\beta N f_N} } : here $X_{N}\to Z_{N}$ 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 Y_N \to -\beta f_N } , 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 f_N } is the free energy density. Let $P_{X_{N}}(x),P_{Y_{N}}(y)$ be the distributions 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 X_N } 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 Y_N } .


• Self-averaging. A random variable $Y_{N}$ is self-averaging when, in the limit 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 } , its distribution concentrates around the average, collapsing to a deterministic value:

  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 \lim_{N \to \infty} Y_N =\lim_{N \to \infty} \overline{Y_N}\equiv \overline{Y_\infty}. }

  This happens when its fluctuations are small compared to the average, meaning that ^[*]

  $\lim _{N\to \infty }{\frac {\overline {Y_{N}^{2}}}{{\overline {Y_{N}}}^{2}}}=1.$

  When the random variable is not self-averaging, it remains distributed in the limit 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 } . When it is self-averaging, sample-to-sample fluctuations are suppressed 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 } is large. This property holds for the free energy of all the disordered systems we will consider. This is very important property: it implies that the free energy (and therefore all the thermodynamics observables, that can be obtained taking derivatives of the free energy) does not fluctuate from sample to sample 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 } is large, and so the physics of the system does not depend on the particular sample. Notice that while intensive quantities like 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 Y_N } (like the free energy density) are self-averaging, quantities scaling exponentially like 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 X_N } (like the partition function) are not necessarily so, see below.



• Average and typical. The typical value of a random variable is the value at which its distribution peaks (it is the most probable value). For self-averaging quantities, in the limit 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 } average and typical value coincide. In general, it might not be so!


We discuss an example to fix the ideas. Often, quantities like 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 Y_N } have a distribution that for 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 } takes the form $P_{Y_{N}}(y)\sim e^{-N^{\omega }g(y)+{\text{subleading}}}$ 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 g(y) } is some positive function 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 \omega>0 } . This is called a large deviation form for the probability distribution, with speed 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^\omega } . This distribution is 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 O(1) } for the value 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 y^{\text{typ}} } such 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 g(y^{\text{typ}})=0 =g'(y^{\text{typ}})} : this value is 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 Y_N } (asymptotically at 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 } ); all the other 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 y } are associated to a probability that 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^\omega} : they are exponentially rare. Consider now an exponentially scaling quantity like 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 X_N = e^{N Y_N} } , and let’s fix 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=1 } . The asymptotic typical values 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 x^{\text{typ}} } 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 y^{\text{typ}} } are related by: 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 y^{\text{typ}} =\lim_{N \to \infty} \frac{\log x^{\text{typ}} }{N}, }

so the scaling 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 x^{\text{typ}} } 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 x^{\text{typ}}\sim e^{N y^{\text{typ}}} } . Let us now look at the scaling of the average. The average 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 X_N } can be computed with the saddle point approximation for 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 } :

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{X_N} =\int dy\, P_{Y_N}(y)\, e^{N y}= \int dy\, e^{N[y- g(y)]+o(N)} =e^{N [y^*-g(y*)]+ o(N) }, }

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 y^* } is the point maximising the shifted function ${\tilde {g}}(y)=y-g(y)$. In this example, 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 y^* \neq y^{\text{ty}} } : the asymptotic of the average 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 X_N } is different from the asymptotic of the typical value. In particular, the average is dominated by rare events, i.e. realisations in which 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 Y_N } takes the value 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 y^*} , whose probability of occurrence is exponentially small.




• Quenched averages. Let us go back 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 X_N } : how to get 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 y^{\text{typ}} } from it? 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 Y_N } is self-averaging, 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 y^{\text{typ}} =\lim_{N \to \infty} \overline{Y_N}= \lim_{N \to \infty} \frac{\overline{\log X_N}}{N} \equiv \lim_{N \to \infty} \frac{{\log x^{\text{typ}}}}{N} }

  where in the last line we have used 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 y^{\text{typ}}= \lim_{N \to \infty} N^{-1} \log x^{\text{typ}}_N } .

  In the language of disordered systems, computing 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 X_N } through the average of its logarithm corresponds to performing a quenched average: from this average, one extracts the correct asymptotic value of the self-averaging quantity 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 Y_N } .


• Annealed averages. The quenched average does not necessarily coincide with the annealed average, 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 y_{\text{a}} = \lim_{N \to \infty} \frac{\log \overline{X_N}}{N}. }

  In fact, it always 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{\log X_N} \leq \log \overline{X_N}} because of the concavity of the logarithm. When the inequality is strict and quenched and annealed averages are not the same, it means 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 X_N } is not self-averaging, and its average value is exponentially larger than the typical value (because the average is dominated by rare events). In this case, to get the correct limit of the self-averaging quantity 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 Y_N } one has to perform the quenched average.^[**] This is what happens in the glassy phases discussed in these TDs.

[*] - See here for a note on the equivalence of these two criteria.

[**] - Notice that the opposite is not true: one can have situations in which the partition function is not self-averaging, but still the quenched free energy coincides with the annealed one.

Problems

Problem 1.1: the energy landscape of the REM

Entropy of the Random Energy Model

The REM has been introduced in ^[1] . In the REM the system can take 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 2^N } configurations 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 \vec \sigma^\alpha=(\sigma^\alpha_1, \cdots, \sigma^\alpha_N)} 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 \sigma^\alpha_i = \pm 1 } . To each configuration 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 \alpha=1, \cdots, 2^N} is assigned a random 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} . The random energies are independent, taken from a Gaussian 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(E) =( 2 \pi N)^{-1/2}e^{-\frac{E^2}{2 N}}.} In this problem we study the random variable 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)dE } , that is the number 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] } . We show that for 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 } it scales 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 \mathcal{N}(E) = e^{N S\left( E/N\right) + o(N)}} . We show that 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 S(\epsilon) } , the quenched entropy density of the model (see sketch), is given by:

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 S(\epsilon)=\begin{cases} \log 2- \frac{\epsilon^2}{2} \quad &\text{ if } |\epsilon| \leq \sqrt{2 \, \log 2} \\ 0 \quad &\text{ if } |\epsilon| >\sqrt{2 \, \log 2} \end{cases} }

The point where the entropy vanishes, 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=- \sqrt{2 \, \log 2} } , is the energy density of the ground state. The entropy is maximal 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 \epsilon=0 } : the highest number of configurations have vanishing energy density.

1. Averages: the annealed entropy. We begin by computing the annealed entropy 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 S_{\text{a}} } , which is defined by the average 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)}= \text{exp}\left(N S_{\text{a}}\left( E/N \right)+ o(N)\right) } . Compute this function using the representation 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)dE= \sum_{\alpha=1}^{2^N} \chi_\alpha(E) dE \;} [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 \chi_\alpha(E)=1} if 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]} 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 \chi_\alpha(E)=0} otherwise]. When does 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 S_{\text{a}} } coincide 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 S } ?

2. Self-averaging. 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 |\epsilon| \leq \sqrt{2 \, \log 2} } the quantity 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} } is self-averaging: its distribution concentrates around the average value 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}} } 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 this 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}} . This property of being self-averaging 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. Rare events. 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 |\epsilon| > \sqrt{2 \, \log 2} } the annealed entropy is negative: 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 the probability to get configurations with those energy 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 } : these configurations are 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}}} ? What is 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} } in this region? Putting everything together, derive the form of the typical value of the entropy density. Why the point where the entropy vanishes coincides with the ground state energy of the model?

Problem 1.2: freezing transition & glassiness

We now compute the equilibrium phase diagram of the model, and in particular the quenched free energy density 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 } which controls the scaling of the typical value of 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 \sim e^{-N \beta \, f +o(N) } } . We show that the free energy 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 f = \begin{cases} &- \left( T \log 2 + \frac{1}{2 T}\right) \quad \text{if} \quad T \geq T_f\\ & - \sqrt{2 \,\log 2} \quad \text{if} \quad T <T_f \end{cases} \quad \quad T_f= \frac{1}{ \sqrt{2 \, \log 2}}. }

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 T_f } a transition occurs, often called freezing transition: in the whole low-temperature phase, the free-energy is “frozen” at the value that it has at the 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= T_f } .

1. The equilibrium transition and the freezing. The partition function the REM 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}. } Using 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} } determined in Problem 1.1, derive the free energy of the model (hint: perform a saddle point calculation). What is the order of this thermodynamic transition?

2. Entropy. What happens to the entropy of the model when the critical temperature is reached, and in the low temperature phase? What does this imply for 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} ?

3. Fluctuations, and back to average vs typical. Similarly to what we did for the entropy, one can define an annealed free 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 f_{\text{a}} } from 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}=e^{- N \beta f_{\text{a}} + o(N)} } : show that in the whole low-temperature phase this is smaller than the quenched free energy obtained above. Putting all the results together, justify why the average of the partition function in the low-T phase is "dominated by rare events".

4. Edwards-Anderson order parameter, and the overlap distribution. The overlap between two configurations 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 \vec{\sigma}^\alpha, \vec{\sigma}^\beta }
   Comment: the low-T phase of the REM is a frozen phase, characterized by the fact that the free energy is temperature independent, and that the typical value of the partition function is very different from the average value. In fact, the low-T phase is also a glass phase where a peculiar symmetry, the so called replica symmetry, is broken. We go back to this concepts in the next sets of problems.
   

   Check out: key concepts

   Self-averaging, average value vs typical value, large deviations, rare events, saddle point approximation, freezing transition.

   To know more

   • Derrida. Random-energy model: limit of a family of disordered models [1]
   • A note on terminology:

   The terms “quenched” and “annealed” come from metallurgy and refer to the procedure in which you cool a very hot piece of metal: a system is quenched if it is cooled very rapidly (istantaneously changing its environment by putting it into cold water, for instance) and has to adjusts to this new fixed environment; annealed if it is cooled slowly, kept in (quasi)equilibrium with its changing environment at all times. Think now at how you compute the free energy, and at disorder as the environment. In the quenched protocol, you compute the average over configurations of the system keeping the disorder (environment) fixed, so the configurations have to adjust to the given disorder. Then you take the log and only afterwards average over the randomness (not even needed, at 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} , if the free-energy is self-averaging). In the annealed protocol instead, the disorder (environment) and the configurations are treated on the same footing and adjust to each others, you average over both simultaneously.