On the Gaussian Perceptron at High Temperature

For σ=(σ i ) i≤N ∈Σ N ={−1,1} N , define $$H\left( \sigma \right) = - \sum\limits_{k \leqslant M} {u\left( {\frac{1}{{\sqrt N }}\sum\limits_{i \leqslant N} {\sigma _i g_i^k } } \right)} ,$$ where (g k i ) i≤N,k≤M are i.i.d. N(0,1), and where u is bounded and Borel measurable. When M is a small proportion α of N, we study the system with random Hamiltonian H, at temperature 1. When α is small enough, we prove that the overlap of two configurations taken independently at random for Gibbs' measure is nearly constant, with a correct estimate of the size of its fluctuations.