One-step replica symmetry breaking of random regular NAE-SAT II

TL;DR

This paper advances the understanding of the random regular k-NAE-SAT model by proving 1RSB properties hold with high probability in the condensation regime, using a novel synthesis of probabilistic and analytical techniques.

Contribution

It introduces a new framework combining small subgraph conditioning and variance decomposition to establish high-probability 1RSB results for the model.

Findings

1RSB properties hold with probability arbitrarily close to one.

Developed a new analytical framework integrating two advanced probabilistic methods.

Overcame technical challenges related to unbounded state spaces in the moment method.

Abstract

Continuing our earlier work in \cite{nss20a}, we study the random regular k-NAE-SAT model in the condensation regime. In \cite{nss20a}, the 1RSB 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.