Eldridge, Forsström, and Schweinhart (2602.22199) represent the Potts lattice Higgs model as a pair of dependent plaquette percolations — translating a quantum field theory object into a geometric one.
The Potts lattice Higgs model assigns spins from Z_q to cells of a cell complex, governed by a Hamiltonian with two terms: a Potts interaction on higher-dimensional cells and an external field term. This is a lattice regularization of gauge-Higgs theory — the kind of object particle physicists use to study confinement and symmetry breaking in controlled settings.
The representation recasts this algebraic object as a geometric one. Instead of spin values and Boltzmann weights, you get two percolation processes — random sets of plaquettes (2-cells) — that are coupled to each other. The coupling encodes the physics: the Higgs field's interaction with the gauge field becomes a correlation structure between which plaquettes are “open” in each percolation.
Why this matters: percolation is one of the best-understood objects in probability theory. Phase transitions in percolation (the emergence of infinite connected clusters) have sharp characterizations. By mapping the Higgs model to percolation, the authors gain access to this machinery. Questions about confinement-deconfinement transitions become questions about percolation thresholds. Questions about the Higgs phase become questions about cluster geometry.
The dependent structure is the hard part. Independent percolation is classical. Dependent percolation — where opening one plaquette changes the probability of opening nearby plaquettes — is substantially harder and is where the physics lives. The gauge-Higgs coupling creates exactly this kind of dependence, and characterizing it requires new tools beyond the standard percolation toolkit.