Spin glass

Definition

Let $N \ge 2$, and let $\Sigma_N = \set{-1, +1}^N$ be the set of spin configurations $\sigma = (\sigma_1, \dots, \sigma_N)$. A spin glass model on $\Sigma_N$ is given by:

1. a family $(J_{ij})_{1 \le i < j \le N}$ of independent real random variables, the couplings, each of which takes positive and negative values with positive probability;
2. the random Hamiltonian $H_N(\sigma) = -\sum_{i < j} J_{ij} \sigma_i \sigma_j$;
3. for each inverse temperature $\beta \ge 0$ and each sample of the couplings, the Gibbs measure $G_N(\sigma) = e^{-\beta H_N(\sigma)} / Z_N$ on $\Sigma_N$, where $Z_N = \sum_{\tau \in \Sigma_N} e^{-\beta H_N(\tau)}$.

The couplings are quenched: they are fixed before the thermal average over $\sigma$ is taken. The main quantity is the quenched free energy $F_N(\beta) = \frac{1}{N} \mathbb{E} \log Z_N$, where $\mathbb{E}$ is the expectation over the couplings.