We present two approximate counting algorithms with $\widetilde{O}(n^{2-c}/\varepsilon^2)$ running time for some constant $c > 0$ and accuracy $\varepsilon$: (1) for the hard-core model when strong spatial mixing (SSM) is sufficiently fast; (2) for spin systems with SSM on planar graphs with quadratic growth, such as $\mathbb{Z}^2$. The latter algorithm also extends to (not necessarily planar) graphs with polynomial growth, such as $\mathbb{Z}^d$, albeit with a running time of the form $\widetilde{O}(n^2\varepsilon^{-2}/2^{c(\log n)^{1/d}})$ for some constant $c > 0$ and $d$ being the exponent of the polynomial growth. Our technique utilizes Weitz's self-avoiding walk tree (STOC, 2006) and the recent marginal sampler of Anand and Jerrum (SIAM J. Comput., 2022).

翻译：我们提出两种近似计数算法，其运行时间为 $\widetilde{O}(n^{2-c}/\varepsilon^2)$，其中 $c > 0$ 为常值，$\varepsilon$ 为精度：(1) 适用于强空间混合（SSM）足够快时的硬核模型；(2) 适用于具有二次增长性的平面图（如 $\mathbb{Z}^2$）上满足 SSM 的自旋系统。第二种算法还可推广至具有多项式增长性的（非平面）图（如 $\mathbb{Z}^d$），但运行时间形如 $\widetilde{O}(n^2\varepsilon^{-2}/2^{c(\log n)^{1/d}})$，其中 $c > 0$ 为常值，$d$ 为多项式增长指数。本技术利用 Weitz 的自回避行走树（STOC, 2006）以及 Anand 和 Jerrum 的最新边际采样方法（SIAM J. Comput., 2022）。