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Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit… — arXiv:2608.21125

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.21125
License (as archived): CC0-1.0
Archived: 2026-10-18T10:39:38.288Z
SHA-256 of the archived text: a855f19f692deefddf492aa1992a90c037a2842f59ce226084023e601bf2062c
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How to cite

Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit… — arXiv:2608.21125 — arXiv (en), https://arxiv.org/abs/2608.21125 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_a397edcf255d (Metraipolis archive dc_a397edcf255d, sha256 a855f19f692deefddf492aa1992a90c037a2842f59ce226084023e601bf2062c)

BibTeX

@misc{dca397edcf255d,
  title = {Two problems for threshold cascades of interacting diffusions on unimodular random trees: front propagation with a Bramson correction, and the continuous-type limit… — arXiv:2608.21125},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.21125},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_a397edcf255d; sha256 a855f19f692deefddf492aa1992a90c037a2842f59ce226084023e601bf2062c},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.21125. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Companion to arXiv: this http URL , which reduces threshold cascades of coupled Ornstein-Uhlenbeck diffusions on graphs converging Benjamini-Schramm to a unimodular Galton-Watson tree, unconditionally in a dissipative regime, to a finite-type Galton-Watson process. Two problems are left open there; we formulate both precisely, supply the analytic framework, and prove partial theorems. First, front propagation. We show the generation-indexed front admits a genuine branching random walk comparison, prove the front depth grows ballistically with an explicit speed $c_*$ given by the Perron root of a tilted mean matrix (unconditional in the dissipative regime), and reduce the conjectured Bramson delay $c_* t - \frac{3}{2c_*}\log t$ to the uniform integrability of a derivative martingale, which we construct at the linearised level. The front central limit theorem and the Bramson correction are stated as conjectures with a proof strategy. Second, the sensitive regime, where the type is a continuous failure strength and the offspring law is a Cox mixture. We prove the mean offspring operator $K$ on $L^2$ of the strength variable is quasi-compact with a spectral gap, the mechanism being Hilbert-Schmidt smoothing of the Ornstein-Uhlenbeck kernel governing strength inheritance. This yields a general-state-space Kesten-Stigum theorem, extends the $n^{-3/2}$ total-progeny law to continuous types with an explicit constant, and gives a strength-resolved central limit theorem. The continuous-type CRT scaling limit is reduced to a multitype invariance principle on a Polish type space, conjectured with all hypotheses verified modulo one tightness estimate. Neither problem is fully closed; each is given a framework, first-order theorems, and a precisely delimited remaining step.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-24 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2608.21125 — 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 2026-10-18T10:38:26.402Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-24?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).