Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension of the failed boundary

Abstract

We study systems of coupled Ornstein-Uhlenbeck diffusions with an absorbing failure state on finite graphs that converge locally weakly to a unimodular Galton-Watson tree. A single failure can trigger a cascade of threshold crossings, and the central question is when the cascade is subcritical or supercritical and what the cluster looks like near criticality. Under an explicitly stated front-decoupling hypothesis (the saturated-coupling regime), verified exactly on the Bethe lattice, the failed cluster of the root converges in law to the genealogy of a finite-type Galton-Watson process whose mean matrix M is computed from a single matrix-valued many-to-one formula. Conditional on this reduction we prove that criticality occurs at rho(M) = 1, that the quenched and annealed growth rates are the tangent at 0 and the value at 1 of a single convex real-analytic pressure phi(beta) = log rho(M(beta)), making the quenched/annealed gap a chord-tangent inequality, that on survival the end boundary of the failed cluster has Hausdorff dimension log rho(M), that at criticality P(|S| = n) ~ C n^{-3/2} and the conditioned cluster rescaled by n^{-1/2} converges to Aldous' Continuum Random Tree under a moment hypothesis, and that the subcritical domain is open and contractible but not convex, with the spectral gap 1 - rho(M) giving quantitative stability. Several natural claims in this setting are shown to be false, including the Hilbert-Schmidt/Krein-Rutman operator programme, the scalar many-to-one identity, and the spectral-radius/dimension identity. The reduction itself is proved unconditionally in the dissipative regime kappa < 1, where a pathwise front-tracking argument shows that back-reaction echoes decay geometrically in graph distance and beat the volume growth of the failed cluster; the strong-coupling regime remains open and is stated as a conjecture.