A novel derivation of the Marchenko-Pastur law through analog bipartite spin-glasses

Abstract

In this work we consider the {\em analog bipartite spin-glass} (or {\em real-valued restricted Boltzmann machine} in a neural network jargon), whose variables (those quenched as well as those dynamical) share standard Gaussian distributions. First, via Guerra's interpolation technique, we express its quenched free energy in terms of the natural order parameters of the theory (namely the self- and two-replica overlaps), then, we re-obtain the same result by using the replica-trick: a mandatory tribute, given the special occasion. Next, we show that the quenched free energy of this model is the functional generator of the moments of the correlation matrix among the weights connecting the two layers of the spin-glass (i.e., the Wishart matrix in random matrix theory or the Hebbian coupling in neural networks): as weights are quenched stochastic variables, this plays as a novel tool to inspect random matrices. In particular, we find that the Stieltjes transform of the spectral density of the correlation matrix is determined by the (replica-symmetric) quenched free energy of the bipartite spin-glass model. In this setup, we re-obtain the Marchenko-Pastur law in a very simple way.