AVERAGING AND FINITE-SIZE ANALYSIS FOR DISORDER - THE HOPFIELD MODEL

A finite-size scaling study of the capacity problem for the Hopfield m odel is presented. Questions of identifying the correct shape of the s caling function, of corrections to finite-size scaling and, in particu lar, the problem of properly dealing with disorder are carefully addre ssed. 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 informatio n about the phase transition can therefore be obtained only by averagi ng their logarithm rather than by considering the logarithm of their a verage - an issue reminiscent of the difference between quenched and a nnealed disorder, but previously ignored in the problem at hand. Our d ata for the Hopfield model yield alpha(c) = 0.141 +/- 0.0015. They are thus closer to the results of a recent one- and two-step replica symm etry 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.