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In this work we consider the {\em analog bipartite spin-glass} (or {\emreal-valued restricted Boltzmann machine} in a neural network jargon), whosevariables (those quenched as well as those dynamical) share standard Gaussiandistributions. First, via Guerra's interpolation technique, we express itsquenched 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 resultby using the replica-trick: a mandatory tribute, given the special occasion.Next, we show that the quenched free energy of this model is the functionalgenerator of the moments of the correlation matrix among the weights connectingthe two layers of the spin-glass (i.e., the Wishart matrix in random matrixtheory or the Hebbian coupling in neural networks): as weights are quenchedstochastic variables, this plays as a novel tool to inspect random matrices. Inparticular, we find that the Stieltjes transform of the spectral density of thecorrelation matrix is determined by the (replica-symmetric) quenched freeenergy of the bipartite spin-glass model. In this setup, we re-obtain theMarchenko-Pastur law in a very simple way.

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