随机正则NAE-SAT的一步复本对称破缺 I
One-step replica symmetry breaking of random regular NAE-SAT I
AI总结：
针对稀疏随机CSP解空间是否符合1RSB凝聚预测的未知问题，本文以随机正则k-NAE-SAT为对象，通过自旋系统矩分析证明其解凝聚于有界数目的解簇、解重叠集中于两值，为1RSB普适类问题提供方法参考。
摘要：
在一大类稀疏随机约束满足问题（CSP）中，统计物理学的深度启发式理论预测，在可满足性阈值之前存在一个由一步复本对称破缺（1RSB）主导的凝聚相变。事实上，在作为此类随机CSP之一的随机正则k-NAE-SAT中，文献[ssz22]已验证其自由能是良定义的，且具体数值符合1RSB预测。然而，对于任何稀疏随机CSP模型，解空间是否确实按照1RSB预测凝聚在O(1)个簇上，这一问题始终未知。本文针对随机正则k-NAE-SAT模型对该问题给出了肯定答案。即我们证明，在概率有界远离零的情况下，大多数解位于数量有界的解簇内，这些解簇的规模与自由能的尺度相当。此外，我们证实两个独立抽取的解之间的重叠精确地集中在两个值上。我们的证明基于对一个自旋系统的详细矩分析，该系统具有编码解簇结构的无限自旋空间。我们相信我们的方法适用于1RSB普适类中的广泛随机CSP。
英文摘要：
In a broad class of sparse random constraint satisfaction problems(CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPs, it was verified \cite{ssz22} that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condenses on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability bounded away from zero, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We believe that our method is applicable to a broad range of random CSPs in the 1RSB universality class.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
