Local Geometry of NAE-SAT Solutions in the Condensation Regime
Name
3618260.3649781.pdf
Size
378.99 KB
Format
Adobe PDF
Checksum (MD5)
76e3dda4f0ea34be39460bd04884ee4c
Author(s)
Sly, Allan
Sohn, Youngtak
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Sly, Allan and Sohn, Youngtak. 2024. "Local Geometry of NAE-SAT Solutions in the Condensation Regime."
Version
Final published version
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.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1145/3618260.3649781
