Approximate Counting for Spin Systems in Sub-Quadratic Time

We present two randomised approximate counting algorithms with widetildeO(n2-c/varepsilon2) running time for some constant c>0 and accuracy varepsilon: (1) for the hard-core model with fugacity λ on graphs with maximum degree Delta when λ=O(Delta-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 mathbbZ2. 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(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 mathbbZd, but with a running time of the form widetildeOleft(n2varepsilon-2/2^c(log n)1/dright) where d is the exponent of the polynomial growth and c>0 is some constant.