Glauber dynamics of the Sherrington--Kirkpatrick spin-glass model: a dynamical cavity method

Abstract

We present a closed theory, exact in the $N\to\infty$ limit, of the equilibrium spin correlations of the Sherrington--Kirkpatrick spin-glass model evolving with Glauber dynamics, above and at the transition temperature $T_c$. The derivation uses a dynamic cavity approach: in the evolution equation for the selected spin the other spins are represented by a fluctuating field. The influence of the selected spin on this field is included at the linear response level; it is given as a sum of the spin's past flips weighted by the correlations of the unperturbed fluctuating field. The self-consistency condition relates the time-dependent correlations of the selected spin and those of the fluctuating field. The theory predicts a $(T-T_c)^{-2}$ divergence of the relaxation time upon approaching the transition and an $At^{-1/2}$ decay of spin correlations at $T_c$. A numerical solution of the theory gives the amplitude, $A=0.964\pm 0.007$. The theoretical predictions are in excellent agreement with Glauber dynamics computer simulations.