An effective-conductance bound for the arboreal gas

The arboreal gas on a finite graph is the random spanning forest in which a forest F has probability proportional to ∏_{e∈F} y_e. We show that on every finite multigraph, two vertices u, v lie in the same tree with probability at most 1/(1+R_eff(u,v)), where R_eff(u,v) is the effective resistance between u and v in the electrical network with conductances y. The proof combines the partition representation of the forest, Kirchhoff's formula relating spanning trees to two-component spanning forests, and Rayleigh monotonicity; the constant is sharp. For a vertex and a ghost vertex joined to every vertex of a discrete torus, the bound is the supersymmetric Mermin–Wagner inequality of Bauerschmidt, Crawford, Helmuth and Swan; here it holds for every pair of vertices on every graph. On ℤ² with weights at most β the bound gives ℙ(0↔x) ≤ 8β/ln(2|x|_∞+1) in every finite volume whose boundary lies at distance at least |x|_∞ from the origin and under every boundary condition, and on the two-dimensional torus of side L ≥ 8 the tree of the origin has expected density at most 9/L + 16β/ln L. Combined with the resampling theorem of Halberstam and Hutchcroft, this gives a proof without supersymmetry that translation-invariant arboreal-gas Gibbs measures on ℤ² have no infinite tree; the supersymmetric method of Bauerschmidt, Crawford, Helmuth and Swan gives the stronger power-law decay. We also prove an exact identity for the connection of a vertex to a set and a lower bound for the separation of two sets, and show that a conjectured exponential set-to-set bound, supported by exact computations on small graphs, would give power-law decay without supersymmetry.