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 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.
Derivation of the model
The REM neglects the correlations between the configurations and assumes the as iid variables.
- Show that the energy distribution is
and determine
The Solution: Part 1
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.
Consider the energies: . they are i.i.d. variables, drawn from the Gaussian distribution . It is useful to use the following notations:
- for . It represents the probability to draw a number smaller than x
- . It represents the probability to draw a number larger than x.
Extreme value stattics for Gaussian variables
We denote
Our goal is to compute the cumulative distribution for large M and iid variables.
We need to understand two key relations:
- the first relation is exact:
- the second relation identifies the typical value of the minimum, namely :
. Hence in the Gaussian case we get:
Close 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 P^>(y) \sim 1/M } . Hence, from 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 \lim_{M\to \infty} (1-\frac{k}{M})^M =\exp(-k)} we re-write the first relation:
Consider 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^>(\tilde y)=\frac 12} . If you draw N independent exponential variables, how many variables (in average) will be greater than ? Repeat the same exercise 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 \tilde \tilde y} 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 P^>( \tilde \tilde y)=\frac 23}
In the large N limit, the distribution becomes 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}
independent.
- Show that in this limit its cumulative takes the from
This is the cumulative distribution of the famous Gumbel distribution.
Let us remark that the precise definition 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 a_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 b_N} fix the mean and the variance of the rescaled 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 \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 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=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 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 a_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 b_N} as 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 N} strongly depend on the particular distributions 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(x)} . We discuss here a family of distribution characterized by a fast decay 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 x}
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 \alpha>0} The key point is to be able to determine 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 A(x)} such that
- 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 p(x) = e^{- x}} shows 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 A(x)=x}
Otherwise 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 A(x)} should be determined asymptotically 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 x}
- 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 A(x)=x^\alpha +(\alpha-1) \log x+...}
- Show that in general 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 A(a_N)= \log N+...} and compute 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 a_N} as 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 \alpha } 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 } .
- Show that the maximum distribution take the form
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 z } Gumbel distributed
- Identify 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 b_N} and discuss its behavior as 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 \alpha }
Number
Bibliography
Bibliography
- Theory of spin glasses, S. F. Edwards and P. W. Anderson, J. Phys. F: Met. Phys. 5 965, 1975