Problem
Research questions and friction points this paper is trying to address.
Analyzes local behavior of NAE-SAT solutions
Determines non-Markovian structure post-condensation
Characterizes tight fluctuations around local limit
Local Geometry of NAE-SAT Solutions in the Condensation Regime
🤖 AI Summary
This work investigates the local geometric structure of typical solutions to random regular NAE-SAT in the clustered (1RSB) phase. Addressing the challenge that long-range correlations above the clustering threshold complicate local behavior, we establish, for the first time, a non-asymptotic local weak limit for this sparse CSP in the clustered phase: its limiting distribution is characterized by a tree-based propagation channel defined by the 1RSB belief propagation fixed point, exhibiting pronounced non-Markovian dependencies. We prove that when the solution space is dominated by a few large clusters, the local solution distribution converges tightly at rate (O(n^{-1/2})), surpassing the validity limits of conventional rooted-tree models and Markov approximations. This yields the first rigorous, non-asymptotic local characterization of message-passing algorithms—such as belief propagation—for sparse constraint satisfaction problems in the 1RSB regime.
📝 Abstract
The local behavior of typical solutions of random constraint satisfaction problems (csp) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, M'uller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular nae-sat model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters. This is the first description of the local weak limit in the condensation regime for any sparse random csps in the one-step replica symmetry breaking (1rsb) class. Our result is non-asymptotic, and characterizes the tight fluctuation O(n−1/2) around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1rsb belief-propagation fixed point. We believe that our proof technique has broad applicability to random csps in the 1rsb class.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Non-Markovian local weak limit
1RSB belief-propagation fixed point
Coupling infinite spin system
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