We present two randomised 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 with fugacity $\lambda$ on graphs with maximum degree $\Delta$ when $\lambda=O(\Delta^{-1.5-c_1})$ where $c_1=c/(2-2c)$; (2) for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as $\mathbb{Z}^2$. For the hard-core model, Weitz's algorithm (STOC, 2006) achieves sub-quadratic running time when correlation decays faster than the neighbourhood growth, namely when $\lambda = o(\Delta^{-2})$. Our first algorithm does not require this property and extends the range where sub-quadratic algorithms exist. Our second algorithm appears to be the first to achieve sub-quadratic running time up to the SSM threshold, albeit on a restricted family of graphs. It also extends to (not necessarily planar) graphs with polynomial growth, such as $\mathbb{Z}^d$, but with a running time of the form $\widetilde{O}\left(n^2\varepsilon^{-2}/2^{c(\log n)^{1/d}}\right)$ where $d$ is the exponent of the polynomial growth and $c>0$ is some constant.

翻译：我们提出了两种随机近似计数算法，其运行时间为 $\widetilde{O}(n^{2-c}/\varepsilon^2)$，其中 $c>0$ 为常数，$\varepsilon$ 为精度要求：(1) 适用于最大度为 $\Delta$ 的图上、逸度为 $\lambda$ 的硬核模型，当 $\lambda=O(\Delta^{-1.5-c_1})$ 且 $c_1=c/(2-2c)$ 时；(2) 适用于具有强空间混合（SSM）性质、且呈二次增长的平面图（如 $\mathbb{Z}^2$）上的自旋系统。对于硬核模型，当关联衰减速度快于邻域增长时（即 $\lambda = o(\Delta^{-2})$ 时），Weitz 算法（STOC, 2006）可实现亚二次运行时间。我们的第一个算法不依赖此性质，从而扩展了亚二次算法存在的参数范围。我们的第二个算法似乎是首个在达到 SSM 阈值时仍能实现亚二次运行时间的算法，尽管限于特定的图族。该算法亦可推广至具有多项式增长（如 $\mathbb{Z}^d$）的（非必须为平面）图，但其运行时间形式为 $\widetilde{O}\left(n^2\varepsilon^{-2}/2^{c(\log n)^{1/d}}\right)$，其中 $d$ 为多项式增长指数，$c>0$ 为某常数。