Path integral diffusion picks a mirror half first, then settles glassy classes in one compound decision
Synopsis
When path integral diffusion (PID) generates a finite Ising spin glass from a point source, each trajectory first chooses one of the two mirror halves of the zero-field measure, then decides among glassy classes inside that half. Theory shows every isotropic PID is Gaussian stochastic localization seen on a different clock, so the protocol sets when decisions happen and what they cost, not what is decided; on exactly enumerable Bethe and SK glasses, nested glassy levels are often settled in one or two distinct events, and these classes appear near the glass transition, keep their identity on cooling, and are absent in a ferromagnet on the same graphs.
Interpretation
An isotropic path integral diffusion started from a point source first commits each trajectory to one of the two mirror halves of the zero-field measure, then to glassy classes inside that half; the per-trajectory order is the same in sparse Bethe and dense SK glasses. Earlier work on diffusion models described a trajectory's commitment to a class as speciation set by the spectrum of the data covariance, but classes and posteriors cannot be computed exactly for real data; spin glasses supply a reference system whose Gibbs measure is known and whose classes can be enumerated exactly. On exactly enumerable Bethe and SK glasses with up to a few dozen spins, every trajectory commits at every level where its branch splits; in the ferromagnet on the same graphs nearly every trajectory makes a single decision, the mirror choice, showing that glassy levels are not a consequence of graph structure or coupling magnitudes alone.
The generator's state can be described by its partial commitments to the target's classes: these probabilities are martingales that move only along class contrasts, and their ensemble average gives the information about each level at a rate set by squared magnetization contrasts. Earlier analyses of diffusion generation stayed at the overall degree of localization or the first speciation time, without a level-by-level information budget. Conditioned on its parent, each split is approximately a binary Gaussian channel whose signal-to-noise ratio is set by the leading eigenvalue of the parent's correlation matrix; the resulting predicted decision times match measured ones on Bethe glassy levels with a median ratio of about 1 and no fitted parameter.
Nested glassy levels are often decided together: a trajectory commits several times on average but forms only one or two distinct events, the mirror choice and a compound glassy decision, which separate further with coupling and size. The earlier picture was either a level-by-level sequence of decisions or a single commitment along the whole path. On Bethe instances with glassy levels, the median ratio of successive glassy commitment times is close to 1, while the mirror choice and the first glassy commitment are at least a factor of about two apart; in a finite-step generator the protocol is felt mainly in the late local stage where individual spins are fixed.
Perspective
The conclusions rest on exactly enumerated systems at finite spin number, plus three selected larger Bethe instances whose posteriors are estimated by sequential Monte Carlo; class identification uses conventional thresholds on mass, overlap gap and collectivity, and the authors make no claim about replica-symmetry breaking, ultrametricity or the thermodynamic limit.
The authors note that classes and posteriors cannot be computed exactly at larger sizes, and the existing larger instances cannot settle whether the variance of glassy splits stays finite with size; the step-allocation recommendation is a tentative extrapolation from three protocols, nine instances and an exact drift, and learned drifts have not been tested.