FRSB in the SK Spin Glass: Convergence to Full-Interval Support at Zero Temperature

Abstract

We study full replica symmetry breaking in the zero-field Sherrington--Kirkpatrick model at finite and zero temperature. At every inverse temperature above the critical point, we prove that the support of the Parisi measure is an interval and that the distance between its upper endpoint and one is at most the reciprocal of the inverse temperature. Consequently, these supports converge in Hausdorff distance to the full overlap interval as the temperature tends to zero. We also prove that the Stieltjes measure associated with the zero-temperature Parisi minimizer has full support after closure at the endpoint one. Both conclusions follow from a common exclusion of constant-order-parameter gaps. The finite-temperature argument uses a de-tilted entrance density and Lopatto's slope-coordinate inequalities, while the zero-temperature argument uses a forward factor, a corresponding slope invariant, and a boundary-layer estimate near the terminal time.

Keywords