Edwards–Anderson model

Example

Let $d \ge 1$ and $L \ge 2$ be integers, and let $\Lambda_L = \set{1, \dots, L}^d \subset \Z^d$. Let $E_L$ be the set of pairs $\set{x, y}$ of points of $\Lambda_L$ at distance $1$. Let $(J_{xy})_{\set{x,y} \in E_L}$ be independent random variables with a common law that is symmetric about $0$, for example standard Gaussian or uniform on $\set{-1, +1}$. The Edwards–Anderson model is the random Hamiltonian $$H_L(\sigma) = -\sum_{\set{x, y} \in E_L} J_{xy} \sigma_x \sigma_y, \qquad \sigma \in \set{-1, +1}^{\Lambda_L},$$ with the Gibbs measure proportional to $e^{-\beta H_L(\sigma)}$ for each sample of the couplings.

Check: the couplings are independent, and a symmetric law that is not concentrated at $0$ gives both signs with positive probability. So the model is a spin glass, with couplings supported on the nearest-neighbour edges of $\Z^d$.