One-Step Replica Symmetry Breaking of Random Regular NAE-SAT II

Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I, arXiv:2011.14270 , 2020), we study the random regular k - nae-sat model in the condensation regime. In Nam et al. (2020), the (1 rsb ) properties of the model were established with positive probability. In this paper, we improve the result to probability arbitrarily close to one. To do so, we introduce a new framework which is the synthesis of two approaches: the small subgraph conditioning and a variance decomposition technique using Doob martingales and discrete Fourier analysis. The main challenge is a delicate integration of the two methods to overcome the difficulty arising from applying the moment method to an unbounded state space.