Spectral Gap of a Kinetically Constrained Spin Model on Trees: Power Law Scaling at Criticality
summary
Kinetically constrained 0–1 spins on a k‑ary tree use product Bernoulli(p) equilibrium and flip a site only when all children are empty; the bootstrap critical p_c equals 1/2 for k=2 (more generally 1/k). Main result: at criticality p = p_c the finite‑depth L spectral gap decays polynomially rather than exponentially, with explicit bounds of the form Gap(L) ≍ L^{-2+o(1)} (proved as L^{-(2+β)} ≤ Gap(L) ≤ L^{-(2-β)} for some β>0 independent of k). Methods combine bootstrap percolation, a multiscale analysis with auxiliary long‑range dynamics and a test‑function upper bound; the approach gives matching polynomial bounds but leaves the sharp exponent open.
AI-written summary · about