Homological percolation in characteristic two: the Ising gauge case

Duncan and Schweinhart proved that the i-dimensional plaquette random-cluster model with cluster weight q on the 2i-dimensional torus has a sharp homological percolation transition at the self-dual point, for coefficients in a field of characteristic other than 2. Their proof needs the point group of the lattice to act irreducibly on the i-th homology of the torus. In characteristic 2 it does not, and the most important case is excluded: q = 2 with F₂ coefficients, which represents the (i−1)-form Ising lattice gauge theory. We prove the theorem for q = 2^r with r ≥ 1 and F₂ coefficients. The representation-theoretic step is replaced by a transfer-matrix inequality, the classical positivity of the partition functions of 't Hooft's electric-flux sectors; read in the random-cluster representation, this inequality compares the probabilities of homological events. As a corollary, for the ferromagnetic (i−1)-form Ising gauge theory on the 2i-dimensional torus (Wegner's model when i = 2), the free energy of every nontrivial 't Hooft twist tends to 0 below the self-dual coupling and to infinity above it as the torus grows. For Bernoulli plaquette percolation (q = 1) the characteristic-2 case remains open for i ≥ 2; we describe obstructions to proving it, and on the 4-torus we prove a computer-assisted divisibility theorem for the classes of integral giant cycles just above p = 1/2.

Corollary B now states the range 0 ≤ J < J_sd covered by its proof; the earlier wording also included J < 0, where the first limit fails for J < −J_sd.

Mathematics Subject Classification (2020): Primary 60K35; Secondary 82B43, 82B20, 82B26, 81T25, 55N10.