LBan-1

In a system with $N$ degrees of freedom, the number of configurations grows exponentially with $N$. For simplicity, consider Ising spins that take two values, ${\sigma }_i=\pm 1$, located on a lattice of size $L$ in $d$ dimensions. In this case, $N=L^d$ and the number of configurations is $M=2^N=e^{N\log 2}$.

In the presence of disorder, the energy associated with a given configuration becomes a random quantity. For instance, in the Edwards-Anderson model:

$E=-\sum _{⟨i,j⟩}{J}_{ij}{\sigma }_i{\sigma }_j,$

where the sum runs over nearest neighbors $⟨i,j⟩$, and the couplings $J_{ij}$ are independent and identically distributed (i.i.d.) Gaussian random variables with zero mean and unit variance.