We study the single-site Glauber dynamics for the fugacity $\lambda$, Hard-core model on the random graph $G(n, d/n)$. We show that for the typical instances of the random graph $G(n,d/n)$ and for fugacity $\lambda < \frac{d^d}{(d-1)^{d+1}}$, the mixing time of Glauber dynamics is $n^{1 + O(1/\log \log n)}$. Our result improves on the recent elegant algorithm in [Bezakova, Galanis, Goldberg Stefankovic; ICALP'22]. The algorithm there is a MCMC based sampling algorithm, but it is not the Glauber dynamics. Our algorithm here is simpler, as we use the classic Glauber dynamics. Furthermore, the bounds on mixing time we prove are smaller than those in Bezakova et al. paper, hence our algorithm is also faster. The main challenge in our proof is handling vertices with unbounded degrees. We provide stronger results with regard the spectral independence via branching values and show that the our Gibbs distributions satisfy the approximate tensorisation of the entropy. We conjecture that the bounds we have here are optimal for $G(n,d/n)$. As corollary of our analysis for the Hard-core model, we also get bounds on the mixing time of the Glauber dynamics for the Monomer-dimer model on $G(n,d/n)$. The bounds we get for this model are slightly better than those we have for the Hard-core model

翻译：我们研究了随机图 $G(n, d/n)$ 上逸度 $\lambda$ 的硬核模型的单点 Glauber 动力学。我们证明，对于随机图 $G(n, d/n)$ 的典型实例和逸度 $\lambda < \frac{d^d}{(d-1)^{d+1}}$，Glauber 动力学的混合时间为 $n^{1 + O(1/\log \log n)}$。我们的结果改进了 [Bezakova, Galanis, Goldberg Stefankovic; ICALP'22] 中近期提出的优雅算法。该算法是一种基于 MCMC 的采样算法，但并非 Glauber 动力学。我们这里的算法更简单，因为我们使用了经典的 Glauber 动力学。此外，我们证明的混合时间上界小于 Bezakova 等人论文中的上界，因此我们的算法也更快。我们证明中的主要挑战是处理具有无界度的顶点。我们通过分支值提供了关于谱独立性的更强结果，并表明我们的吉布斯分布满足熵的近似张量化。我们推测这里给出的上界对于 $G(n, d/n)$ 是最优的。作为硬核模型分析的推论，我们还得到了 $G(n, d/n)$ 上单体-二聚体模型的 Glauber 动力学混合时间的上界。我们对该模型得到的上界稍优于硬核模型的上界。