L-1

Spin glass Transition

Experiments

Parlare dei campioni di rame dopati con il magnesio, marino o no: trovare due figure una di suscettivita e una di calore specifico, prova della transizione termodinamica.

Edwards Anderson model

We consider for simplicity the Ising version of this model.

Ising spins takes two values $\sigma =\pm 1$ and live on a lattice of $N$ sitees $i=1,2,\ldots ,N$. The enregy is writteen as a sum between the nearest neighbours <i,j>:

$E=-\sum _{<i,j>}J_{ij}\sigma _{i}\sigma _{j}$

Edwards and Anderson proposed to study this model for couplings $J$ that are i.i.d. random variables with zero mean. We set $\pi (J)$ the coupling distribution indicate the avergage over the couplings called disorder average, with an overline:

${\bar {J}}\equiv \int dJ\,J\,\pi (J)=0$

It is crucial to assume ${\bar {J}}=0$, otherwise the model displays ferro/antiferro order. We sill discuss two distributions:

• Gaussian couplings: $\pi (J)=\exp \left(-J^{2}/2\right)/{\sqrt {2\pi }}$
• Coin toss couplings, $J=\pm 1$, selected with probability $1/2$.

The SK model

Sherrington and Kirkpatrik considered the fully connected version of the model with Gaussian couplings:

$E=-\sum _{i,j}{\frac {J_{ij}}{2N}}\sigma _{i}\sigma _{j}$

At the inverse temperature $\beta$, the partion function of the model is

$Z=\sum _{\alpha =1}^{2^{N}}z_{\alpha },\quad {\text{with}}\;z_{\alpha }=e^{-\beta E_{\alpha }}$

Here $E_{\alpha }$ is the energy associated to the configuration $\alpha$. This model presents a thermodynamic transition at $\beta _{c}=??$.

Random energy model

The solution of the SK is difficult. To make progress we first study the radnom energy model (REM) introduced by B. Derrida.

Derivation of the model

The REM neglects the correlations between the $2^{N}$ configurations and assumes the $E_{\alpha }$ as iid variables.

• Show that the energy distribution is

$p(E_{\alpha })={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {E_{\alpha }^{2}}{2\sigma ^{2}}}}$

and determine $\sigma ^{2}$

The Solution

Extreme value stattics for Gaussian variables

Generically, finding the distribution of the maximum of a set of random variables is a non-trivial problem, which appears in many contexts ranging from the maximal height of water in a river to fluctuations in stock markets We consider N independent random variables $(x_{1},...,x_{N})$ drawn from the same distribution $p(x)$. We denote

$y_{N}=\min(x_{1},...,x_{N})$

It is useful to use the following notations for the cumulative distributions

$P^{<}(x)=\int _{-\infty }^{x}dx'p(x')\qquad \qquad P^{>}(x)=\int _{x}^{+\infty }dx'p(x')$

Let us denote by $q_{N}(y)$ the distribution of $y_{N}$ and by $Q_{N}(y)={\text{Prob}}(y_{N}<y)$ its cumulative distribution.

• Write $Q_{N}(y)$ in terms of $P^{<}(y)$. (Help: Start to write this relation for $N=2,3,...$).

This is the fundamental relation of Extreme statistics and we analyze its consequences in the large N limit where, analogously to the central limit theorem, extremes statistics display universal features.

• In particular shows that in the large N limit we can write

$Q_{N}(y)\sim \exp \left(-NP^{>}(y)\right)$

Consider now the case $\lambda =1$

• Write $P^{>}(x)$ and $P^{<}(x)$. (Remember that $x$ is a positive number.)
• Write $Q_{N}(y)$ and $q_{N}(y)$.
• Plot $q_{N}(y)$ for different values of N.

We want now to give a natural definition for the number $a_{N}$ and $b_{N}$.

Consider $P^{>}({\tilde {y}})={\frac {1}{2}}$. If you draw N independent exponential variables, how many variables (in average) will be greater than ${\tilde {y}}$? Repeat the same exercise with ${\tilde {\tilde {y}}}$ such that $P^{>}({\tilde {\tilde {y}}})={\frac {2}{3}}$

• Justify that $a_{N}$ can be estimated from

$P^{>}(a_{N})={\frac {1}{N}}$

• Compute $a_{N}$ for the exponential distribution and justify that

$Q_{N}\left(y=a_{N}+z\right)$

In the large N limit, the distribution $\pi (z)$ becomes $N$ independent.

• Show that in this limit its cumulative takes the from

$\Pi (z)=e^{-e^{-z}}$ $\Pi (z)=e^{-e^{-z}}$

This is the cumulative distribution of the famous Gumbel distribution.

Let us remark that the precise definition of $a_{N}$ and $b_{N}$ fix the mean and the variance of the rescaled distribution $\pi (z)$ At variance with the central limit case the mean will be different from zero and the variance different from one.

• Compute the mean, the variance and the asymptotic behavior of the Gumbel distribution. Draw the distribution. Explain why $z=0$ is a special point

Generic case: Universality of the Gumbel distribution

The Gumbel distribution is the limit distribution of the maxima of a large class of function. We can say that the Gumbel distribution plays, for extreme statistics, the same role of the Gaussian distribution for the central limit theorem.

By contrast the behavior of $a_{N}$ and $b_{N}$ as a function of $N$ strongly depend on the particular distributions $p(x)$. We discuss here a family of distribution characterized by a fast decay for large $x$

$p(x)\sim ce^{-x^{\alpha }}$

where $\alpha >0$ The key point is to be able to determine $A(x)$ such that

$P^{>}(x)=\exp(-A(x))$

• For $p(x)=e^{-x}$ shows $A(x)=x$

Otherwise $A(x)$ should be determined asymptotically for large $x$

• Show that $A(x)=x^{\alpha }+(\alpha -1)\log x+...$
• Show that in general $A(a_{N})=\log N+...$ and compute $a_{N}$ as a function of $\alpha$ for large $N$.
• Show that the maximum distribution take the form

$\lim _{N\to \infty }Q_{N}(y)=\left(y=a_{N}+{\frac {z}{A'(a_{N})}}\right)$

with $z$ Gumbel distributed

• Identify $b_{N}$ and discuss its behavior as a function of $\alpha$

Bibliography

• Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975