LBan-1

In a system of $N$ degrees of freedom we have a number of configuration which is exponential in $N$. For simplicity consider the Ising spins that take two values, $\sigma _{i}=\pm 1$, located on a lattice of size $L$ in dimension $d$, $N=L^{d}$ and the number of configuration is $M=2^{N}=e^{N\log 2}$ configurations. In presence of disorder the energy associated to a given configuration is random. For instance, in the Edwards Anderson model:

$E=-\sum _{\langle i,j\rangle }J_{ij}\sigma _{i}\sigma _{j}.$

here the sum in intended over nearest neighbors $\langle i,j\rangle$ and the couplings $J_{ij}$ that are independent and identically distributed (i.i.d.) Gaussian variables with a zero mean and unit variance.

If we consider $N_{trial}$

Since the systems are random, the quantities that describe their properties (the free energy, the number of configurations of the system that satisfy a certain property, the magnetization etc) are also random variables, with a distribution.