Approximate Counting for Spin Systems in Sub-Quadratic Time

We present two randomised approximate counting algorithms with O^e(n^2−c/ε²) running time for some constant c > 0 and accuracy ε: 1. for the hard-core model with fugacity λ on graphs with maximum degree ∆ when λ = O(∆^−1.5−c¹) where c1 = c/(2 − 2c); 2. for spin systems with strong spatial mixing (SSM) on planar graphs with quadratic growth, such as Z². 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(∆⁻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 Z^d, but with a running time of the form O (n²ε⁻²/2^{c(log n)1/d)} where d is the exponent of the polynomial growth and c > 0 is some constant.