Approximating the total variation distance between spin systems

Spin systems form an important class of undirected graphical models. For two Gibbs distributions μ and nu induced by two spin systems on the same graph G = (V, E), we study the problem of approximating the total variation distance dTV(μ,nu) with an ε-relative error. We propose a new reduction that connects the problem of approximating the TV-distance to sampling and approximate counting. Our applications include the hardcore model and the antiferromagnetic Ising model in the uniqueness regime, the ferromagnetic Ising model, and the general Ising model satisfying the spectral condition. Additionally, we explore the computational complexity of approximating the total variation distance dTV(μS,nuS) between two marginal distributions on an arbitrary subset S subseteq V. We prove that this problem remains hard even when both μ and nu admit polynomial-time sampling and approximate counting algorithms.