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| * The famous Gumbel distribution: | | * The famous Gumbel distribution: |
| <center><math>Q_M(\epsilon) \sim \exp\left(-e^{ y_N (\epsilon-a_M)}\right) </math> </center> | | <center><math>Q_M(\epsilon) \sim \exp\left(-e^{ y_N (\epsilon-a_M)}\right) </math> </center> |
| * The typical location of the minimum around <math> a_M </math> | | * the typical fluctuations of the minimum <math> \sim 1/y_N</math>. In particular we can write: |
| * the typical fluctuations of the minimum (i.e. its standard deviation) <math> \sim 1/y_N \sim N^{\omega}</math>
| | <math> \overline{E_{\min}^2} - \overline {E_{\min}}^2 \propto N^{2\omega}</math> |
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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
and live on a lattice of
sitees
.
The enregy is writteen as a sum between the nearest neighbours <i,j>:
Edwards and Anderson proposed to study this model for couplings
that are i.i.d. random variables with zero mean.
We set
the coupling distribution indicate the avergage over the couplings called disorder average, with an overline:
It is crucial to assume
, otherwise the model displays ferro/antiferro order. We sill discuss two distributions:
- Gaussian couplings:

- Coin toss couplings,
, selected with probability
.
Edwards Anderson order parameter
The SK model
Sherrington and Kirkpatrik considered the fully connected version of the model with Gaussian couplings:
At the inverse temperature
, the partion function of the model is
Here
is the energy associated to the configuration
.
This model presents a thermodynamic transition at
.
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.
This model neglects the correlations between the
configurations and assumes the
as iid variables.
- Show that the energy distribution is
and determine
We provide different solutions of the Random Energy Model (REM). The first one focus on the statistics of the smallest energies among the ones associated to the
configurations. For this, we need to become familiar with the main results of extreme value statistic of iid variables.
Extreme value statistics
Consider the
energies:
as iid variables, drawn from the distribution
(Gaussian, but we remain general in this section). It is useful to use the following notations:
, it is the probability to find an energy smaller than E.
, it is the probability to find an energy larger than E.
We denote
Our goal is to compute the cumulative distribution
for large M. To achieve this we need two key relations:
- The first relation is exact:
- The second relation identifies the typical value of the minimum, namely
:
.
Let us consider the limit,
, which allow to re-write the first relation:
This relation holds only when
and one hase to expand around this value.
However, a direct Taylor expansion does not ensures that probabilities remain positive. Hence, we define
and remark that
is a decreasing function. We propose the following Taylor expansion
Depending on the distribution
we have a different dependence on N or M of both
. It is convenient to define
We will see that three different scenarios occur depending on the sign of
. Using this expansion we derive:
- The famous Gumbel distribution:
- the typical fluctuations of the minimum
. In particular we can write:
Density of states above the minimum
For a given disorder realization, we compute
, the number of configurations above the minimum with an energy smaller than
. The key relation for this quantity is:
Taking the average
, we derive
Now, in the integral
is the energy of the minimum, hence we can use
Exercise L1-A: the Gaussian case
Specify these results to the Guassian case and find
for
- the typical value of the minimum
%
- The expression

- The expression of the Gumbel distribution for the Gaussian case
Bibliography
- Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975