Small random perturbation of a classical mean field model (Q1593623)

The thermodynamic limit of a classical mean field system perturbated by a small Sherrington-Kirkpatrick term is derived. The classical mean field system is a spin system ruled by the Hamiltonian \[ H_N^f(\sigma) = N \cdot f \left ({1\over N} \sum _{i=1}^N \sigma _i\right) \] where \(\sigma =(\sigma _1,\ldots ,\sigma _N)\in [-1,1]^N\) is the configuration of the system, and \(f:[-1,1]\to R\) is a smooth function. Let \(P_\sigma \) denote the product measure \(\rho ^{\otimes N}\), where \(\rho \) is the single spin distribution on \([-1,1]\), and \(E_\sigma \) the expectation with respect to \(P_\sigma \). Then the asymptotic behaviour of the partition function \( Z_{N,\beta}^f = E_\sigma \exp \beta H_N^f(\sigma) \) and the Gibbs distribution \[ G_{N,\beta}^f (d\sigma _1,\ldots ,d\sigma _N) = {\exp \beta H_N^f(\sigma)\over Z_{N,\beta}^f} P_\sigma (d\sigma _1,\ldots ,d\sigma _N) \] for \(N\to \infty \) can be studied. The results are known. In the present paper the Hamiltonian \( H_N^{f,\alpha}(\sigma) = H_N^f(\sigma) + N^{-\alpha /2} H_N^{SK}(\sigma) \) is considered, where \(\alpha \in (1/2, 1)\) and the perturbation term \[ H_N^{SK}(\sigma) = N^{-1/2} \sum _{1\leq i < j\leq N} J_{ij} \sigma _j \sigma _j \] is the Sherrington-Kirkpatrick (random) Hamiltonian with a system \((J_{ij})_{\{1\leq i<j\leq N\}}\) of i.i.d. random variables with common standard Gaussian \({\mathcal N}(0,1)\) distribution. The main result of the paper is the investigation of the asymptotic behaviour (which differs substantially from the above basic mean field model) of the (random) partition function \(Z_{N,\beta}^{f,\alpha}\) and the (random) Gibbs distribution \(G_{N,\beta}^{f,\alpha}\).