Sharp RS/RSB Boundary and the Almeida–Thouless Line in the SK Model

Determine whether the Almeida–Thouless stability condition, defined by the fixed-point q solving q = E[tanh^2(β√q Z + h)] together with the inequality β^2 E[sech^4(β√q Z + h)] ≤ 1 (with Z standard normal), exactly characterizes the replica-symmetric region of the Sherrington–Kirkpatrick model for all inverse temperatures β and external fields h; equivalently, identify the precise replica-symmetric/replica-symmetry-breaking phase boundary of the Sherrington–Kirkpatrick model.

Background

The replica-symmetric (RS) regime of the Sherrington–Kirkpatrick (SK) model is associated with concentration of the overlap around a deterministic value q that solves a self-consistency equation, while the Almeida–Thouless (AT) criterion provides a stability condition involving q. Establishing the exact RS/RSB boundary requires proving whether the AT condition is both necessary and sufficient for replica symmetry in SK.

The chapter notes several partial results related to the AT line (e.g., work of Guerra, Chen, and Brennecke–Yau), but emphasizes that confirming the AT condition as the exact boundary remains unresolved and is identified as a central open problem.