Sequential Dynamics in Ising Spin Glasses

Abstract

We present the first exact asymptotic characterization of sequential dynamics for a broad class of local update algorithms on the Sherrington-Kirkpatrick (SK) model with Ising spins. Focusing on dynamics implemented via systematic scan -- encompassing Glauber updates at any temperature -- we analyze the regime where the number of spin updates scales linearly with system size. Our main result provides a description of the spin-field trajectories as the unique solution to a system of integro-difference equations derived via Dynamical Mean Field Theory (DMFT) applied to a novel block approximation. This framework captures the time evolution of macroscopic observables such as energy and overlap, and is numerically tractable. Our equations serve as a discrete-spin sequential-update analogue of the celebrated Cugliandolo-Kurchan equations for spherical spin glasses, resolving a long-standing gap in the theory of Ising spin glass dynamics. Beyond their intrinsic theoretical interest, our results establish a foundation for analyzing a wide variety of asynchronous dynamics on the hypercube and offer new avenues for studying algorithmic limitations of local heuristics in disordered systems.