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Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension… — arXiv:2608.15987

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.15987
License (as archived): CC0-1.0
Archived: 2026-10-18T20:25:11.128Z
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Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension… — arXiv:2608.15987 — arXiv (en), https://arxiv.org/abs/2608.15987 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_a77dc72071bd (Metraipolis archive dc_a77dc72071bd, sha256 53844d8bc77031709b4893bc94e0de1c21ec9241a015ae0114921548078e5e79)

BibTeX

@misc{dca77dc72071bd,
  title = {Threshold cascades of interacting diffusions on unimodular random trees: a conditional branching-process universality theorem, the annealed pressure, and the dimension… — arXiv:2608.15987},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.15987},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_a77dc72071bd; sha256 53844d8bc77031709b4893bc94e0de1c21ec9241a015ae0114921548078e5e79},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.15987. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-18 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.15987 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-18T19:45:36.382Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-18?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).