The fluctuations of the overlap in the Hopfield model with finitely many patterns at the critical temperature (Q1960924)

The authors consider the Hopfield model with a finite number of patterns at the critical temperature \(\beta=1\). It is known that the so-called overlap parameter, \(m_N\), converges to zero under the Gibbs measure almost surely when the system size, \(N\), tends to infinity. The authors prove a non-standard limit theorem for the blown up overlap, \(N^{1/4} m_N\), in the sense that this quantity converges in law to some random variable whose probability density is random with respect to the quenched disorder and explicitly given in terms of a function of some family of Gaussian random variables.