On the Gaussian perceptron at high temperature (Q1610858)

The author studies a modification of the perceptron model. In fact, he considers a Hamiltonian function \[ H(\sigma)=-\sum_{k\leq M} u\left(\frac 1{\sqrt N}\sum_{i\leq N} \text{s}_ig^k_i\right) \] where \(\{g^k_i\}\) is a family of i.i.d. normal random variables and \(u\) is a bounded measurable function. It is proven that the Gibbs measure \(\mu_N(\text{s}) \equiv \exp(-H(\text{s}))/Z\) concentrates on configurations with fixed relative overlap \(q\), where \(q\) is characterized as the solution of some nonlinear fix point equation which are known from the replica theory developed in the theoretical physics literature.