For a large class of random constraint satisfaction problems (CSP), deep but non-rigorous theory from statistical physics predict the location of the sharp satisfiability transition. The works of Ding, Sly, Sun (2014, 2016) and Coja-Oghlan, Panagiotou (2014) established the satisfiability threshold for random regular $k$-NAE-SAT, random $k$-SAT, and random regular $k$-SAT for large enough $k\geq k_0$ where $k_0$ is a large non-explicit constant. Establishing the same for small values of $k\geq 3$ remains an important open problem in the study of random CSPs. In this work, we study two closely related models of random CSPs, namely the $2$-coloring on random $d$-regular $k$-uniform hypergraphs and the random $d$-regular $k$-NAE-SAT model. For every $k\geq 3$, we prove that there is an explicit $d_{\ast}(k)$ which gives a satisfiability upper bound for both of the models. Our upper bound $d_{\ast}(k)$ for $k\geq 3$ matches the prediction from statistical physics for the hypergraph $2$-coloring by Dall'Asta, Ramezanpour, Zecchina (2008), thus conjectured to be sharp. Moreover, $d_{\ast}(k)$ coincides with the satisfiability threshold of random regular $k$-NAE-SAT for large enough $k\geq k_0$ by Ding, Sly, Sun (2014).

翻译：对于一大类随机约束满足问题（CSP），统计物理中的深刻但非严格理论预测了尖锐可满足性转变的位置。Ding、Sly、Sun（2014, 2016）以及Coja-Oghlan、Panagiotou（2014）的工作建立了随机正则$k$-NAE-SAT、随机$k$-SAT以及随机正则$k$-SAT的可满足性阈值，其中$k$足够大时$k\geq k_0$，且$k_0$是一个较大的非显式常数。对于较小的$k\geq 3$建立相同结果仍然是随机CSP研究中的一个重要开放问题。本文研究随机CSP的两个密切相关的模型，即随机$d$-正则$k$-一致超图上的$2$-着色模型和随机$d$-正则$k$-NAE-SAT模型。对于每个$k\geq 3$，我们证明存在一个显式的$d_{\ast}(k)$，它为这两个模型提供了可满足性上界。在$k\geq 3$时，我们的上界$d_{\ast}(k)$与Dall'Asta、Ramezanpour、Zecchina（2008）关于超图$2$-着色的统计物理预测一致，因此被推测为精确的。此外，对于足够大的$k\geq k_0$，$d_{\ast}(k)$与Ding、Sly、Sun（2014）关于随机正则$k$-NAE-SAT的可满足性阈值一致。