TejasviTomar
October 2026 · Preprint

Homological percolation in characteristic two: the Ising gauge case

Tejasvi Singh Tomar

In plain words

Percolation asks when random links first join up across a whole system. Its version for surfaces asks when random patches first wrap around a four-dimensional torus. Duncan and Schweinhart, building on earlier work with Kahle, found exactly when this happens for a family of such models. Their proof does not cover the case of arithmetic modulo 2, which keeps track only of whether numbers are even or odd. This paper settles that case for part of the family, including the Ising gauge theory from physics. It follows that, as the system grows, the cost of each ’t Hooft twist in that theory tends to zero for couplings below Wegner’s self-dual coupling and grows without bound for couplings above it.

Abstract

Duncan and Schweinhart proved that the i-dimensional plaquette random-cluster model on the 2i-dimensional torus has a sharp homological percolation transition at the self-dual point p_sd = √q/(1+√q). Their proof assumes that the homology coefficient field does not have characteristic 2, because it uses the irreducibility of H_i(T^{2i}; F) under the point group of the lattice. The excluded case contains the most important example, q = 2 with F₂ coefficients, which represents the (i−1)-form Ising lattice gauge theory. We prove the theorem for q = 2^r and F = F₂, for every r ≥ 1, by replacing the representation-theoretic step with a transfer-matrix inequality, the classical positivity of the partition functions of 't Hooft's electric-flux sectors. As a corollary, every 't Hooft twist free energy of the four-dimensional Ising gauge theory tends to 0 below Wegner's self-dual coupling and to infinity above it. For Bernoulli plaquette percolation (q = 1) on the 4-torus we prove that near p = 1/2 the classes of the integral giant cycles are divisible by any fixed integer M. The proof adapts the enhancement argument of Aizenman and Grimmett. Its local step, which converts a pivotal plaquette into an enhanceable one, rests on a finite certificate for a capsule-shaped region of the lattice, checked by an exact computation and in Lean, and on an induction over the length of that region. For q = 1 the sharp threshold with F₂ coefficients remains open; we reduce it to a single inequality, which holds in all cases we enumerated, and show that the threshold lies in [√2 − 1, 2 − √2].

A 32-by-32 grid with periodic edges in which most open edges belong to one large teal cluster, and a thick magenta loop runs from the top edge to the bottom edge, wrapping around the torus.
Illustration of the lowest case, i = 1, of the paper's Theorem A: a sample of the random-cluster model with q = 2 on a 32 × 32 torus at the self-dual point p = √2/(1+√2), drawn by computer. One cluster's edges are teal, and its magenta cycle wraps around the torus: a giant cycle. For i = 1 the theorem was known; the paper's new case is i ≥ 2, where edges become i-dimensional plaquettes on the 2i-dimensional torus.