Averaging and finite-size analysis for disorder: The Hopfield model

A finite-size scaling study of the capacity problem for the Hopfield model is presented. Questions of identifying the correct shape of the scaling function, of corrections to finite-size scaling and, in particular, the problem of properly dealing with disorder are carefully addressed. At first-order phase transitions, like the one considered here, relevant physical quantities typically scale exponentially with system size, and it is argued that in disordered systems reliable information about the phase transition can therefore be obtained only by averaging their logarithm rather than by considering the logarithm of their average — an issue reminiscent of the difference between quenched and annealed disorder, but previously ignored in the problem at hand. Our data for the Hopfield model yield αc = 0.141 ± 0.0015. They are thus closer to the results of a recent one- and two-step replica symmetry breaking (RSB) analysis, and disagree with that of an earlier one-step RSB study, with those of previous simulations, and with that of a recent paper using an infinite-step RSB scheme.