How Path Integral Diffusion Decides a Finite Spin Glass

원문 초록

When an AI diffusion-type generator samples a frustrated spin glass, which macroscopic alternatives does it decide first, and when? We study this for bridge-diffusion models of the Path Integral Diffusion (PID) type started from a point source. We show that every isotropic PID is a time-changed Gaussian stochastic localization, and conversely every signal-to-noise schedule is realized by a PID protocol. What a continuous-time generator decides therefore does not depend on the protocol, which only sets when and at what cost. We describe the generator's state by its partial commitments: the probabilities it assigns, at each moment, to nested classes of configurations fixed by the target. The partial commitments are martingales that move only along class contrasts. Their ensemble average gives the information about each level at an exact rate set by squared magnetization contrasts. Each split is decided, approximately, like a binary Gaussian channel at signal-to-noise ratio 1.04/\lambda_P, with \lambda_P the leading eigenvalue of the parent's correlation matrix. The theory is applied to Bethe-lattice and Sherrington--Kirkpatrick (SK) glasses with up to 64 spins. It is observed generically that a trajectory first shifts its weight to one of the two mirror halves of the zero-field measure and then to the glassy classes inside it. These classes appear near the glass transition and keep their identity on cooling. Nested glassy levels have similar \lambda_P and are decided together, so a trajectory commits in one or two distinct events---the mirror choice and a compound glassy decision---which separate further with coupling and size, as the spectra predict. For a generator that takes finitely many steps, the protocol matters through the late, local stage rather than the macroscopic decisions.