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

Abstract

Continuing our earlier work in Nam et al. (One-step replica symmetry breaking of random regular NAE-SAT I, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" ext-link-type="uri" xlink:href="http://arxiv.org/abs/2011.14270">arXiv:2011.14270</jats:ext-link>, 2020), we study the random regular <jats:italic>k</jats:italic>-<jats:sc>nae-sat</jats:sc> model in the condensation regime. In Nam et al. (2020), the (1<jats:sc>rsb</jats:sc>) 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.