On the Marchenko–Pastur law in analog bipartite spin-glasses**The authors are pleased to dedicate this paper to Giorgio Parisi on the occasion of his seventieth birthday.

Abstract

In recent decades, statistical mechanics of disordered systems (mainly spin-glasses) has become one of the main tools to investigate complex systems, probably due to the celebrated Replica Symmetry Breaking scheme of Parisi Theory and its deep implications.In this work we consider the analog bipartite spin-glass (or 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 matrix whose entries are the correlation coefficients between 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 acts as a powerful 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.