Approximate Counting for Spin Systems in Sub-Quadratic Time

TL;DR

The paper tackles sub-quadratic time randomised approximation for partition functions of spin systems, focusing on the hard-core model with small fugacity on bounded-degree graphs and on planar graphs with strong spatial mixing. It introduces two algorithms achieving $\widetilde{O}(n^{2-c}/\varepsilon^{2})$ running times for some $c>0$, with the first relaxing correlation-decay requirements relative to Weitz’s method and the second approaching the SSM threshold on planar graphs (and extending to polynomial-growth graphs) with a near-quadratic bound that further slows only by a subpolynomial factor in growth. The approach combines Weitz’s self-avoiding walk tree with Anand–Jerrum’s lazy marginal sampler to estimate marginals efficiently, enabling a quadratic speedup in marginal estimation and, consequently, sub-quadratic counting for both hard-core and more general 2-spin systems. These results broaden the landscape of scalable approximate counting for spin systems, providing sub-quadratic algorithms on planar and polynomial-growth graph families, and offering practical implications for estimating partition functions in statistical physics and combinatorial counting problems.

Abstract

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 $λ$ on graphs with maximum degree $Δ$ when $λ=O(Δ^{-1.5-c_1})$ 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 $λ= o(Δ^{-2})$. 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.

Figures (3)

• Figure 4: Running time comparison among MCMC, Weitz's algorithm, and \ref{['small-lambda']}
• Figure 5: Circle packings of some lattices. (a): $\mathbb{Z}^2$ grid, $R=1$. (b): Kisrhombille tiling, $R=2-\sqrt{3}$. (c): degree-$3$ Bethe lattice, $R=0$.
• Figure 6: The graph $G(n)$ when $n$ is even. The number next to a vertex indicates its distance from the starting vertex $S$.

Theorems & Definitions (26)

• Theorem 1.1
• remark 1.2: Decay rate vs. neighbourhood growth
• Theorem 1.3
• Theorem 1.4
• definition 2.1
• definition 2.2
• definition 2.3: SSM
• Theorem 2.4: Theorem 3.1 of Wei06
• Lemma 3.1
• Lemma 3.2