Percolation probability in the thermodynamic limit at zero string tension for negative chemical potential

Establish that, in the classical two-dimensional Z2 lattice gauge theory on the square lattice with grand-canonical Hamiltonian H = − h ∑⟨i,j⟩ τ^x⟨i,j⟩ − μ ∑_j n_j subject to Z2 Gauss’s law, the percolation probability ρ—defined as the probability that a cluster of bonds with τ^x = −1 spans opposite edges of the lattice—satisfies ρ(L, β h = 0, β μ < 0) → 1 as the system size L → ∞.

Background

The paper introduces a real-space renormalization group (RG) scheme that uses the percolation probability of Z2 electric strings as an order parameter for confinement in a classical 2D Z2 lattice gauge theory with matter. The authors analytically derive the confinement phase diagram and support their RG predictions with Monte Carlo simulations.

In the regime of zero string tension (β h = 0), they present numerical evidence that the percolation probability increases with system size for negative chemical potential β μ < 0. They note that at β μ = 0 the system sits at the Bernoulli percolation threshold, where crossing probabilities approach 1/2 in finite-size scaling. Motivated by Kolmogorov’s zero-one law, they conjecture that for β μ < 0 the percolation probability in the thermodynamic limit equals 1.