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

Abstract: Companion to arXiv:2608.XXXXX, 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 $c_<em>$ given by the Perron root of a tilted mean matrix (unconditional in the dissipative regime), and reduce the conjectured Bramson delay $c_</em> t - \frac{3}{2c_*}\log t$ 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 $L<sup>2$ 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 $n<sup>{-3/2}$ 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.

• The paper presents a law of large numbers for front propagation, proving that the velocity almost surely converges to a critical speed c_* in the dissipative regime.
• The analysis outlines a framework for deriving second-order behavior, including a Bramson correction, contingent on verifying the uniform integrability of a derivative martingale.
• It introduces a Kesten–Stigum theorem for continuous-type offspring and relates to Aldous' Continuum Random Tree, emphasizing the need for additional estimates to validate the conjecture.

Setting and dependence on the companion paper

This paper addresses two problems left open by a companion study of threshold cascades of coupled Ornstein–Uhlenbeck (OU) diffusions on finite graphs converging in the Benjamini–Schramm sense to a unimodular Galton–Watson tree $T \sim \mathrm{UGW}(g)$. The companion paper reduces the cascade — unconditionally in the dissipative regime $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$ — to a finite-type Galton–Watson process with mean matrix $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$, where $m_* = g''(1)/g'(1)$ is the size-biased forward-degree mean. The present work takes two of the resulting open problems, supplies each with an analytic framework, and proves first-order theorems while stating the remaining second-order steps as precisely delimited conjectures.

Two black-box results from the companion paper are used throughout: (R1) front decoupling, which identifies the failed cluster up to total-variation error with the genealogy of the multitype Galton–Watson process with mean matrix $M$ on the dissipative region; and (R2) the sensitive-regime mean operator $K$, a positive integral operator on the continuous failure-strength variable whose criticality $\rho(K)=1$ was previously only squeezed between finite matrices. All unconditional statements inherit the regime $\{\kappa < 1\}$ and the iterated limit $n \to \infty$, $\delta \to \infty$; nothing conditional in the companion paper is re-opened here.

The front as a branching random walk

The first part concerns front propagation. Each failed vertex $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$0 carries a depth $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$1 and a type $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$2; by (R1) the collection of depths of generation-$$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$3 failed vertices is the $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$4-th generation of a branching random walk (BRW). Because every edge advances depth by exactly one, the generation-indexed speed is trivially $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$5; the nontrivial object is the time-indexed front

$$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$6

where edge transmission times $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$7 are i.i.d.\ given endpoint types, by the saturated-regime independence established in the companion paper. Thus $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$8 is exactly the maximal-displacement process of a BRW on the genealogical tree with positive i.i.d.\ edge weights.

The speed is defined via the activation Laplace transform $$\kappa = \gamma_{\max} L_\sigma \Delta_{\max}/\alpha_{\min} < 1$$9 and the tilted mean matrix $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$0:

$M_{d,d'} = m_*\, q(d')\, p_{d,d'}$1

the standard Biggins velocity variational formula, with optimiser $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$2. This is the branching-random-walk / Fisher–KPP linear selection principle: $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$3 is fixed by the leading edge of the front, where the population is sparse and dynamics linearised. The paper notes that the front-energy supermartingale of the companion paper is a type-weighted version of the Biggins additive martingale evaluated along the front, which ties the two treatments together.

Ballistic front propagation

The main unconditional result of Part I is a law of large numbers for the front. Under (R1), dissipativity $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$4, supercriticality $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$5, irreducibility of $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$6, and a finite exponential moment for the activation law — which holds for the OU first-passage time under uniform ellipticity, since its tails are exponential — the paper proves that almost surely on survival,

$M_{d,d'} = m_*\, q(d')\, p_{d,d'}$7

The proof reduces to the classical maximal-displacement LLN for supercritical irreducible multitype BRW with exponentially integrable increments: the minimal time to reach depth $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$8 satisfies $M_{d,d'} = m_*\, q(d')\, p_{d,d'}$9, and monotone inversion gives the front speed. This is a linear first-order theorem, and it is unconditional within the dissipative regime modulo the standing dependence on (R1).

Second order: the derivative martingale and the Bramson delay

The second-order behaviour is where the paper is deliberately incomplete. For branching Brownian motion, Bramson's celebrated result gives a maximum of $m_* = g''(1)/g'(1)$0, with tightness and limiting law later obtained via the derivative martingale. The paper constructs the analogous objects: with $m_* = g''(1)/g'(1)$1 the left Perron eigenvector of $m_* = g''(1)/g'(1)$2,

$m_* = g''(1)/g'(1)$3

A proposition establishes that $m_* = g''(1)/g'(1)$4 is a nonnegative martingale for all $m_* = g''(1)/g'(1)$5 in the domain, that $m_* = g''(1)/g'(1)$6 is a signed martingale, that $m_* = g''(1)/g'(1)$7 a.s.\ at the critical tilt, and that $m_* = g''(1)/g'(1)$8 a.s.\ with $m_* = g''(1)/g'(1)$9 on survival provided uniform integrability holds.

The entire Bramson programme is then reduced to a single estimate: uniform integrability of $M$0, which by the $M$1 criterion reduces to a checkable $M$2-moment condition on the degree law against the activation tail. The paper states as a conjecture that this yields

$M$3

with a randomly-shifted Gumbel decorated point process limit, the shift being a multiple of $M$4. Three ingredients are identified: the spine change of measure at $M$5 (available now), the $M$6 moment (automatic for bounded degrees), and the Bramson barrier argument. In the bounded-degree dissipative regime, ingredient (2) is immediate since the one-generation weight is bounded by $M$7 times a bounded tilted weight; the authors state plainly that the only reason this is not claimed as a theorem there is that the decorated-point-process convergence has not been transcribed from Euclidean BRW to the genealogical-tree setting.

One genuinely open question is flagged as high-value: whether quenched disorder in the UGW environment modifies the $M$8 coefficient itself. The pressure function of the companion paper already shows quenched and annealed speeds can differ off boundary-homogeneous trees, and analogues in BRW in random environment do exhibit shifted constants. Whether such a shift occurs here is not mere transcription and remains open.

Quasi-compactness of the sensitive-regime mean operator

Part II turns to the sensitive regime, where the vertex type is the continuous failure strength $M$9 (the overshoot past threshold) and offspring follow a Cox mixture governed by a strength kernel $K$0 inherited from OU forced-response dynamics. The mean offspring operator acts on $K$1, with $K$2 the Gaussian reference measure, as

$K$3

with criticality $K$4.

The central structural theorem asserts that under a Gaussian-type density for $K$5, uniform ellipticity $K$6, and Wasserstein-2 Lipschitz dependence of the forcing, three conclusions hold. First, $K$7 is Hilbert–Schmidt, hence compact, so its nonzero spectrum is discrete. Second, $K$8 is positivity-improving and irreducible, so Jentzsch's theorem gives a simple spectral radius with strictly positive eigenfunction $K$9 strictly dominating the rest of the spectrum — a genuine spectral gap. Third, the Doob $\rho(K)=1$0-transform $\rho(K)=1$1 is a Markov operator with a spectral gap, hence uniformly ergodic with unique invariant law $\rho(K)=1$2 and exponential mixing.

The analytic mechanism is worth emphasising: the Hilbert–Schmidt norm is finite because the Gaussian forced-response density lies in $\rho(K)=1$3. The paper explicitly contrasts this with the boundary operator of the companion paper, which is not Hilbert–Schmidt because the singular tree Martin kernel admits no smoothing. The earlier pessimism about the sensitive regime therefore concerned only the boundary object; the strength operator enjoys one of the smoothest kernels in probability, and compactness, spectral gap, and Perron theory come without additional cost. The finite-matrix squeeze of the companion paper is thus revealed as the criticality shadow of a full gap-endowed spectral theory rather than a ceiling.

Continuous-type Kesten–Stigum theory and the $\rho(K)=1$4 law

With the spectral gap in hand, the paper proves a general-state-space Kesten–Stigum theorem. Let $\rho(K)=1$5 be the empirical strength measure of generation $\rho(K)=1$6. Then $\rho(K)=1$7 is a nonnegative martingale; under an $\rho(K)=1$8 condition — automatic under uniform ellipticity and finite degree variance, reducing to $\rho(K)=1$9 — it converges a.s.\ and in $\{\kappa < 1\}$0 with the dichotomy $\{\kappa < 1\}$1. Under finite offspring second moments the convergence upgrades to $\{\kappa < 1\}$2: the second-moment recursion closes because the tensorised operator $\{\kappa < 1\}$3 inherits the spectral gap from $\{\kappa < 1\}$4. Moreover, the normalised empirical law converges a.s.\ weakly to the deterministic tilted stationary law $\{\kappa < 1\}$5, independently of the surviving realisation, via the ergodic theorem for the spine chain.

At criticality $\{\kappa < 1\}$6, a corollary extends the total-progeny law to continuous types:

$\{\kappa < 1\}$7

where $\{\kappa < 1\}$8 is the offspring-count variance averaged over the stationary spine strength $\{\kappa < 1\}$9. The exponent $n \to \infty$0 is unchanged from the finite-type case; only the constant changes, now computed against $n \to \infty$1. The proof conditions on the strength process and applies the Otter–Dwass formula plus a local CLT to the resulting Cox-mixed single-type critical Galton–Watson process.

Strength-resolved CLT and the CRT scaling limit

In the supercritical regime, the paper proves a spatial central limit theorem: for centred bounded observables $n \to \infty$2 with $n \to \infty$3, conditionally on survival,

$n \to \infty$4

where $n \to \infty$5 is a Green–Kubo sum, finite by the spectral gap. The fluctuation scale is $n \to \infty$6, mixed by the Kesten–Stigum limit $n \to \infty$7; higher cumulants are negligible by the branching-process CLT machinery.

The continuous-type CRT scaling limit is stated as a conjecture: at criticality, with exponential offspring moments, the cluster conditioned on $n \to \infty$8, rescaled by $n \to \infty$9, should converge in the Gromov–Hausdorff–Prokhorov topology to Aldous' Continuum Random Tree with an explicit constant built from $\delta \to \infty$0 and $\delta \to \infty$1. Two proof routes are laid out — mesh discretisation with uniform-in-mesh tightness control, or direct invocation of infinitely-many-types invariance principles for Polish type spaces — and the missing input in either route is named precisely: a single uniform tightness/second-moment estimate that has not been written out. The statement is accordingly a delimited conjecture, not an open-ended problem.

Limitations and open questions

The paper is explicit about what remains unproved. On the front side, the Bramson delay and decorated-point-process limit rest on the uniform-integrability conjecture; even in the bounded-degree regime where the moment condition is automatic, the transcription of the decorated-point-process convergence from Euclidean BRW to genealogical trees is outstanding. Whether UGW disorder shifts the $\delta \to \infty$2 coefficient is entirely open. On the sensitive-regime side, the CRT limit requires the missing uniform tightness estimate, and the spectral-gap theorem depends on hypotheses — Gaussian-type kernel density, uniform ellipticity, Wasserstein Lipschitz forcing — whose verification for specific cascade parameters is not carried out here. All unconditional statements additionally inherit the dissipative-regime restriction and the black-box status of (R1) from the companion paper.

Conclusion

The paper converts two open problems into structured research programmes with rigorous partial results. For front propagation, ballistic growth with explicit variational speed $\delta \to \infty$3 is proved unconditionally in the dissipative regime, and the conjectured Bramson correction $\delta \to \infty$4 is reduced to a single derivative-martingale estimate. For the sensitive regime, Hilbert–Schmidt smoothing of the OU strength kernel yields quasi-compactness and a spectral gap, from which a general-state-space Kesten–Stigum theorem, the persistence of the $\delta \to \infty$5 progeny law with an explicit continuous-type constant, and a strength-resolved CLT all follow. What is proved is proved unconditionally modulo the companion framework; what is conjectured comes with its missing step named.