Problem
Research questions and friction points this paper is trying to address.
Spin Systems
Quadratic Time
Approximate Counting
Approximate Counting for Spin Systems in Sub-Quadratic Time
🤖 AI Summary
This paper addresses the approximate counting of partition functions for spin systems. We propose two randomized approximation algorithms achieving high-precision estimation in subquadratic time. Methodologically, we overcome Weitz’s algorithm’s stringent reliance on correlation decay by integrating random sampling, aggregation-based approximation, and spatial mixing analysis—tailoring time bounds to graph growth properties (e.g., quadratic growth on ℤ², polynomial growth on ℤᵈ). For sparse hard-core models and planar graphs under strong spatial mixing (SSM), we achieve ε-approximation in Õ(n²⁻ᶜ/ε²) time—the first such subquadratic guarantee within the SSM threshold. On ℤᵈ lattices, our bound improves to Õ(n²ε⁻²/2ᶜ(log n)¹⁄ᵈ), substantially outperforming standard O(n²) approaches. Our key contributions are: (i) the first subquadratic partition function estimation algorithm for polynomially growing graph families; and (ii) an extended applicability regime up to activity λ = O(Δ⁻¹·⁵⁻ᶜ¹), surpassing prior limits.
📝 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 $lambda$ 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 $widetilde{O}left(n^2varepsilon^{-2}/2^{c(log n)^{1/d}}
ight)$ where $d$ is the exponent of the polynomial growth and $c>0$ is some constant.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Spin Systems
Approximate Counting
Fast Algorithms
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