T-I

Goal: derive the equilibrium phase diagram of the simplest spin-glass model, the Random Energy Model (REM). The REM is defined assigning 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} of the system 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}}} .

Key concepts: average value vs typical value, self-averaging quantities, rare events, freezing transition, saddle-point.

Some probability notions relevant at large N

Let us introduce a few concepts that will be useful for the problems below.

• We will consider 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 $N\gg 1$ 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 O(e^{N}) } ; the scaling 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 $O(1)$ 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 } . A standard example are 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, $Z\sim e^{-\beta Nf}$: here 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 \to Z } 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 } , 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 } is the free energy.


• Let 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_N(x), P_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 $Y_{N}$. In general, 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 $P_{N}(y)\sim e^{-Ng(y)+o(N)}$ 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 g(y) } is some positive function. This is called a large deviation form for the probability distribution. 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 $y_{\text{ty}}$ 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{ty}})=0 } : 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 } ; 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} : they are exponentially rare .
• Averages over distributions having a large deviation form can usually 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 } . If $f(Y_{N})$ is a function 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 } which scales slower that exponential 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 } , then ${\overline {f(Y_{N})}}=\int dyP_{N}(y)f(y)=\int dxe^{-Ng(y)+o(N)}f(y)\sim f(y_{\text{ty}})$

  because the integral is dominated by the region 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)=0 } , since all the other contributions are exponentially suppressed.

• We define 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 } as $x_{\text{ty}}=e^{Ny_{\text{ty}}}$. This is also equivalent 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_{\text{ty}} =e^{ \overline{\log X_N}}} since 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}=\int dx P_N(x) \log x = N \int dy P_N(y) \, y= N \int dye^{-N g(y)+o(N)} \, y \sim N y_{\text{ty}} }

  Notice that while the average value of $Y_{N}$ coincides with the typical value (choose 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(y)=y} in the formula above), this is in general not the case for quantities growing exponentially fast 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 } : the average value of these exponentially scaling quantities is in general much larger than the typical value.

• A random variable depending on a parameter $N$ is self-averaging when the width of its distribution goes to zero 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 N \to \infty } . 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 } .

Problem 1.1: the energy landscape of the REM

Entropy of the Random Energy Model

In this problem we study the random variable ${\mathcal {N}}(E)dE$, 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 } , its typical value scales as

${\mathcal {N}}(E)=e^{N\Sigma \left({\frac {E}{N}}\right)+o(N)},\quad \quad \Sigma (\epsilon )={\begin{cases}\log 2-\epsilon ^{2}\quad &{\text{ if }}|\epsilon |\leq {\sqrt {\log 2}}\\0\quad &{\text{ if }}|\epsilon |>{\sqrt {\log 2}}\end{cases}}$

The function 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(\epsilon) } is the entropy of the model (see sketch). The point where it 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{\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 \Sigma_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 \Sigma_a\left( \frac{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 $\Sigma _{a}$ coincide with the entropy defined above (which we define as the “quenched” entropy in the following)?

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{\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 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 \sigma^2= \overline{\mathcal{N}^2}- \overline{\mathcal{N}}^2 \sim \overline{\mathcal{N}}} ; by the central limit theorem, 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 \mathcal{N} } is self-averaging 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 |\epsilon|< \sqrt{\log 2} } . 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. Rare events vs typical values. 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{\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? Justify why the point where the entropy vanishes coincides with the ground state energy of the model.

Comment: this analysis of the landscape suggests that in the 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 } limit, the fluctuations due to the randomness become relevant when one looks at the bottom of their energy landscape, close to the ground state energy. We show below that this intuition is correct, and corresponds to 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 } has an interesting behaviour at low temperature.

Problem 1.2: the free energy and the freezing transition

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}{4 T}\right) \quad \text{if} \quad T \geq T_c\\ & - \sqrt{\log 2} \quad \text{if} \quad T <T_c \end{cases} \quad \quad T_c= \frac{1}{2 \sqrt{\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_c } 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_c } .

1. The thermodynamical 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_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_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".

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 : it is a phase where a peculiar symmetry, the so called replica symmetry, is broken. We go back to this concepts in the next sets of problems.