Morphology of frozen labyrinths from irreversible threshold dynamics

Abstract

Majority threshold dynamics, in which each agent adopts the dominant state in a weighted neighborhood, relaxes a binary field toward consensus or stripes. We study what happens when this rule is made irreversible: each agent, interacting through a Gaussian kernel on a lattice, may flip out of its local weighted minority at most once. The reversible form is threshold dynamics of Merriman-Bence-Osher type, an exactly solvable calibration in which interfaces move by mean curvature with closed-form lattice pinning and mobility. Irreversibility changes the outcome. From random initial conditions, the one-flip rule freezes balanced non-consensus labyrinths that reversible relaxation drives away. The patterned regime is a window of initial spin compositions around equal balance, narrowing as the interaction range grows, controlled by a standardized bias whose onset is independent of scale over a fourfold range. The frozen morphology is arrested coarsening: bicontinuous at balance, with a feature width that grows sublinearly and falls below the interaction range at large scales. We determine the mechanism by intervention. Fixing the initial condition while varying the update order shows that the coarse domain layout is deterministic, while the update order enters only in a secondary first-passage race that fine-tunes the wall positions the deterministic dynamics has already set. Those walls are marked by frustration: agents frozen against their own local field. This signature is identically absent from any reversible relaxation, is overwhelmingly interfacial, and carries almost none of the pattern's large-scale shape.