Spin-glasses are natural Gibbs distributions that have been studied in Theoretical CS for many decades. Recently, they have been gaining attention from the community as they emerge naturally in neural computation and learning, network inference, optimisation and other areas. We study the problem of efficiently sampling from spin-glass distributions when the underlying graph is a typical instance of $G(n,d/n)$, i.e., the random graph on $n$ vertices such that each edge appears independently with probability $d/n$, and $d=Θ(1)$. Our focus is on the 2-spin model at inverse temperature $β$. We consider this distribution to be one of the most interesting case of spin-glasses, and one of the most challenging to analyse, since its Gaussian couplings give rise to unbounded interaction. We employ the well-known Glauber dynamics to sample from the aforementioned distribution. We show that for the typical instances of the 2-spin model on $G(n,d/n)$, the mixing time of Glauber dynamics is $O\left(n^{1+\frac{25}{\sqrt{\log d}}}\right)$, for any $β<\frac{1}{4\sqrt{d}}$. Our results can also be adapted for the case of spin-glass distributions with bounded interactions. In that respect, we obtain rapid mixing of Glauber dynamics for the Viana-Bray model on $G(n,d/n)$ when $β<\frac{1}{4\sqrt{d}}$. This improves on the current best bound which is $β<\frac{0.18}{\sqrt{d}}$. We utilise stochastic localisation, and in particular, we build and improve on the scheme introduced in [Liu, Mohanty, Rajaraman and Wu: FOCS 2024]. This is the first time that stochastic localisation is used for diluted spin-glasses, where both degrees and interactions can be unbounded.

翻译：自旋玻璃是理论计算机科学领域研究数十年的自然吉布斯分布。近年来，由于它们自然出现在神经计算与学习、网络推理、优化等领域，日益受到学界关注。我们研究当底层图是$G(n,d/n)$的典型实例（即n个顶点上每条边以概率$d/n$独立出现的随机图，其中$d=Θ(1)$）时，从自旋玻璃分布中高效采样的问题。我们聚焦于逆温度$β$下的2-自旋模型。由于该分布的高斯耦合导致无界相互作用，我们将其视为自旋玻璃中最有趣且最具分析挑战性的情形之一。我们采用经典的格劳伯动力学从上述分布中采样。我们证明：对于$G(n,d/n)$上2-自旋模型的典型实例，当$β<\frac{1}{4\sqrt{d}}$时，格劳伯动力学的混合时间为$O\left(n^{1+\frac{25}{\sqrt{\log d}}}\right)$。我们的结果也可适用于有界相互作用的自旋玻璃分布。在此方面，我们得到当$β<\frac{1}{4\sqrt{d}}$时，$G(n,d/n)$上Viana-Bray模型的格劳伯动力学快速混合，这改进了当前最佳界$β<\frac{0.18}{\sqrt{d}}$。我们利用随机局域化方法，特别是构建并改进了[Liu, Mohanty, Rajaraman and Wu: FOCS 2024]中引入的方案。这是首次将随机局域化应用于度与相互作用均可无界的稀疏自旋玻璃。