Sherrington–Kirkpatrick model

Example

Let $N \ge 2$ and let $(g_{ij})_{1 \le i < j \le N}$ be independent standard Gaussian random variables. The Sherrington–Kirkpatrick model is the random Hamiltonian $$H_N(\sigma) = -\frac{1}{\sqrt{N}} \sum_{1 \le i < j \le N} g_{ij} \sigma_i \sigma_j, \qquad \sigma \in \set{-1, +1}^N,$$ with partition function $Z_N(\beta) = \sum_{\sigma} e^{-\beta H_N(\sigma)}$ for the inverse temperature $\beta \ge 0$.

Check: the couplings $J_{ij} = g_{ij} / \sqrt{N}$ are independent and symmetric, so each takes both signs with probability $1/2$. So it is a spin glass in which every spin interacts with every other spin. For each fixed $\sigma$, $H_N(\sigma)$ is a centred Gaussian variable of variance $\frac{1}{N} \binom{N}{2} = \frac{N-1}{2}$, so the energy grows like $N$, as it should.