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.

• The paper conditionally reduces threshold cascades of interacting Ornstein–Uhlenbeck diffusions to a finite-type branching process with mean matrix M, proving survival occurs precisely when its Perron root satisfies ρ(M)>1.
• It shows that the annealed growth rate ρ(M) generally exceeds the quenched geometric rate, with a convex pressure function quantifying the Jensen gap and the failed boundary dimension given by dim_H(∂S)=log ρ(M).
• At criticality, the failed-cluster size follows the universal n^-3/2 law and, under exponential-moment assumptions, converges after rescaling to the Continuum Random Tree, while unconditional reduction is established in a dissipative regime defined by κ<1 and κρ(M)<1.

This paper develops a rigorous theory of threshold-triggered failure cascades for systems of coupled Ornstein–Uhlenbeck diffusions on finite graphs converging in the Benjamini–Schramm sense to a unimodular Galton–Watson tree $T\sim UGW(g)$. Its central contributions are a conditional reduction of the cascade to a finite-type branching process; a spectral identity identifying criticality with the Perron root of an explicitly computed mean matrix; a convex pressure function that organizes the quenched/annealed gap; a Hausdorff-dimension theorem for the boundary of the failed cluster; mean-field universality of the total-progeny exponent and, under an exponential-moment hypothesis, of the Continuum Random Tree (CRT) shape; and an unconditional proof of the reduction in an explicit dissipative parameter regime. The manuscript is also deliberately cautionary: several natural claims are proved false, and the paper is explicit that its main theorems are conditional on a reduction hypothesis that is proved only on the Bethe lattice, in part in general, and unconditionally in the dissipative regime $\kappa<1$.

The model and the load-bearing reduction

Vertices carry independent hazard-domain types $d\in D$, $|D|=m$, and evolve according to coupled SDEs with linear dissipation, logistic inter-vertex coupling, and an absorbing failure state. The cascade initiated by failing the root produces a random cluster $S\subset V(T)$ of eventually failed vertices. The key analytic object is the directed pairwise first-passage weight: for a failed parent forcing a child, the transmission probability $p_{d,d'}$ is given exactly, for the linearized two-barrier OU problem, by a ratio of Gaussian integrals (a Wald-type hitting-probability formula), and is uniformly bounded below by uniform ellipticity.

The paper isolates a saturated-coupling regime: once a vertex fails, its coupling output is pinned near $1$ by a margin $\delta$, so the forcing a child sees is parent-trajectory-independent to within $\varepsilon(\delta)$. Under this regime the paper posits Hypothesis 1 (front-decoupling): the law of the failed cluster is within total variation $\eta_n\downarrow 0$ of the genealogy of a multitype Galton–Watson process with type set $\kappa<1$0 and mean matrix

$\kappa<1$1

where $\kappa<1$2 is the size-biased mean forward degree (the branching number of $\kappa<1$3, a normalization the paper stresses is distinct from the root offspring mean $\kappa<1$4). Theorem 3.1 then identifies criticality: survival is positive iff $\kappa<1$5. The paper is careful to state that Sections 4–8 are theorems about the limiting branching process, and their bearing on the SDE is exactly as strong as this hypothesis.

A structural dichotomy governs when the finite-type collapse is legitimate. In the saturated regime, a child's transmission depends on the parent only through the failure event, so failure strengths are conditionally i.i.d. and the process is finite-type. Outside saturation, the failure strength $\kappa<1$6 is inherited by children with positive covariance, the offspring law becomes a Cox mixture on the continuous type space $\kappa<1$7, and criticality is governed by a Perron root of an integral operator. The scalar hidden in the latter regime — the failure-strength variance — is precisely the $\kappa<1$8 quantity that defeats the Girsanov approach.

The spectral/dimension gap and the pressure function

A central corrective result (Theorem 4.1) concerns two competing growth rates. The quenched, geometric-mean rate $\kappa<1$9, obtainable by Birkhoff's theorem along $d\in D$0-typical rays, equals the harmonic-measure-dimension expression one would naively equate with the spectral radius. The annealed criticality is instead governed by $d\in D$1. The gap is a Jensen gap: $d\in D$2, with equality if and only if $d\in D$3 is constant on $d\in D$4. Thus the tempting identity $d\in D$5 holds on boundary-homogeneous trees (in particular the Bethe lattice) and strictly undercounts criticality otherwise. The paper's diagnosis is conceptual: a Perron root is an arithmetic average of multiplicative weights, while a harmonic-measure dimension is a geometric average; the two coincide only without fluctuations.

This dichotomy is completed by a one-parameter structure. Defining tilted mean matrices $d\in D$6 and the pressure $d\in D$7, Theorem 5.1 shows $d\in D$8 is convex and real-analytic, with $d\in D$9 and $|D|=m$0; the quenched/annealed gap is exactly the chord–tangent inequality, and the equality case follows from the identification of $|D|=m$1 with the Green–Kubo variance along the Perron-tilted spine chain. The proof rests on the matrix many-to-one identity

$|D|=m$2

whose exponential growth rate is $|D|=m$3. The paper proves the scalar form of this identity is false in general, because consecutive edge factors along a ray share a vertex type; the scalar expression is valid only when $|D|=m$4 is a left eigenvector of $|D|=m$5, and $|D|=m$6 and $|D|=m$7 are not comparable in a fixed direction.

Dimension of the failed boundary

Extending the finite-dimensional collapse past boundary-blind observables, Theorem 5.4 proves, conditionally on the reduction and for $|D|=m$8, that almost surely on survival the end boundary $|D|=m$9 of the failed cluster satisfies

$S\subset V(T)$0

The upper bound is a cylinder-covering estimate against the expected type-counts; the lower bound constructs, via the multitype Kesten–Stigum martingale (whose $S\subset V(T)$1 convergence uses the finite offspring variance), a random measure on $S\subset V(T)$2 whose $S\subset V(T)$3-energy is finite for $S\subset V(T)$4. The corollary is that the failed boundary has strictly positive codimension $S\subset V(T)$5 in $S\subset V(T)$6 unless transmission is perfect. This is a boundary-resolved quantity computed entirely by the finite matrix $S\subset V(T)$7; the paper notes that what genuinely requires infinite-dimensional structure is the fluctuation theory on $S\subset V(T)$8 — the multifractal spectrum of transmission-weighted flows, governed by the Legendre transform $S\subset V(T)$9, and the harmonic measure of the cluster, where the dimension-drop phenomenon should reappear.

Universality: the $p_{d,d'}$0 exponent and the CRT

At criticality $p_{d,d'}$1, Theorem 6.2 establishes, via the Otter–Dwass formula and a one-dimensional local central limit theorem, that

$p_{d,d'}$2

with off-critical correction $p_{d,d'}$3, $p_{d,d'}$4. The exponent depends only on criticality and finiteness of the offspring variance — not on drifts, thresholds, or boundary structure — and the proof requires no general-state-space Kesten–Stigum theory, because $p_{d,d'}$5 is a count blind to the boundary mark. The offspring variance is computed explicitly: a Bernoulli-thinning term plus a degree-fluctuation term, both finite under $p_{d,d'}$6; in the sensitive regime an additional Cox term appears, still finite.

Under an additional exponential-moment hypothesis (inherited from Miermont's invariance principle for multitype Galton–Watson trees, and flagged by the author as the one place demanding more than a second moment), Theorem 6.4 upgrades the exponent to a shape: conditionally on $p_{d,d'}$7, the cluster rescaled by $p_{d,d'}$8 converges in the Gromov–Hausdorff–Prokhorov topology to Aldous' Continuum Random Tree, with an explicit scale factor $p_{d,d'}$9 computed from the second-moment matrices and Perron data of $1$0. The author expects the finite-variance case to hold and cites available relaxations, but prefers to state the theorem under hypotheses for which a complete citation exists.

Dissolution of the infinite-dimensional operator programme

A natural programme — replacing the finite next-generation matrices by a bounded operator $1$1 on $1$2 and invoking Kreĭn–Rutman theory — is shown to be both unnecessary and generically impossible. Unnecessary: in the saturated regime every spectral question (simplicity, monotonicity, real-analyticity, the van den Driessche–Watmough lift unifying $1$3 with vanishing spectral abscissa of the linearized drift) is a textbook finite-dimensional Perron–Frobenius fact. Impossible: the limiting boundary kernel inherits the diagonal singularity of the tree Martin kernel, and a shell-counting argument shows the Hilbert–Schmidt energy integral behaves as $1$4, which diverges for essentially all supercritical offspring laws. The paper's verdict is blunt: this is not a hard lemma awaiting proof but a false target to be abandoned; $1$5 must instead be obtained as $1$6 via the matrix many-to-one identity.

Stability of the subcritical domain

Theorem 8.1 establishes the geometry of the parameter domains. The subcritical domain $1$7 is open, with smooth critical hypersurface $1$8 where the gradient is nonzero; the supercritical domain $1$9 is the convex object (a superlevel set of the log-concave transmission probability, whose log-concavity is proved in an appendix), while $\delta$0 is its non-convex but contractible complement, star-shaped toward the zero-coupling locus by an explicit coupling-retraction homotopy. The paper explicitly flags the invalid inference — "sublevel sets of a concave function are convex" — as a false direction. On $\delta$1 the resolvent $\delta$2 is a bounded nonnegative matrix, expected cluster size is $\delta$3 and real-analytic, and the spectral gap $\delta$4 gives quantitative perturbative stability. The paper also notes there is no topological invariant to protect: the failed cluster is a.s. a contractible subforest at every parameter, so the genuine near-critical phenomenon is analytic (resolvent blow-up, critical slowing-down).

The Bethe lattice serves as the exact verification: there the reduction holds exactly, $\delta$5, the pressure is affine and the Jensen gap vanishes, the critical cluster has explicit constant $\delta$6 in the $\delta$7 law, and the supercritical failed boundary has dimension $\delta$8 — an exponentially thin bundle of rays of relative codimension $\delta$9. The paper notes this homogeneity is precisely why Bethe-lattice checks cannot detect the general gap.

Small-noise asymptotics

The paper corrects a tempting small-noise claim. Laplace's method on the exact Wald representation yields

$\varepsilon(\delta)$0

an exponential rate times an algebraic prefactor $\varepsilon(\delta)$1, not the multiplicative $\varepsilon(\delta)$2 form. The prefactor is irrelevant for the criticality threshold and the $\varepsilon(\delta)$3 exponent (which depend on $\varepsilon(\delta)$4 and variance finiteness) and enters only the constant in the total-progeny law; the main results use the exact Wald formula throughout.

Unconditional front decoupling in the dissipative regime

The honest core of the paper is its treatment of the reduction hypothesis. The naive Girsanov route is proved to fail: the back-reaction drift is $\varepsilon(\delta)$5 on the cascade timescale, the quadratic-variation term is $\varepsilon(\delta)$6, and the Radon–Nikodym density is not tight toward $\varepsilon(\delta)$7. Cellwise, the paper proves the saturated-regime ingredients (single-cell correctness to $\varepsilon(\delta)$8, sibling-decoupling, vanishing short-cycle corrections), but these do not control cross-generation error compounding over the exponential volume of the tree.

The positive result is a pathwise mechanism. Two structural facts make the argument work: echo sources live on the failed cluster rather than the ambient tree (non-failed vertices exert only $\varepsilon(\delta)$9 coupling), and OU dissipation localizes echoes in space and time. With the dissipation ratio

$\eta_n\downarrow 0$0

a synchronous-coupling cone-of-influence estimate (weighted Gronwall in a norm adapted to the echo sources) shows influence decays geometrically at rate $\eta_n\downarrow 0$1 in graph distance. When $\eta_n\downarrow 0$2 and $\eta_n\downarrow 0$3, echo decay strictly beats volume growth — the relevant volume is the failed cluster, whose expectation is $\eta_n\downarrow 0$4-bounded, not $\eta_n\downarrow 0$5 — and an exploration telescoping yields the quantitative total-variation bound

$\eta_n\downarrow 0$6

Consequently all main theorems hold unconditionally, as statements about the SDE, on the open parameter region $\eta_n\downarrow 0$7. At criticality the bound degrades to $\eta_n\downarrow 0$8 conditionally on $\eta_n\downarrow 0$9, so the critical limit theorems hold unconditionally only along joint limits with $\kappa<1$00. The paper flags, rather than proves, the non-vacuousness subtlety that spontaneous crossings at moderate noise remain negligible on a strict open subset of $\kappa<1$01. The front-energy quantity built from the left Perron vector is shown to be a supermartingale up to bounded sources, giving a precise sense in which the front is asymptotically Markovian; no singular-SPDE machinery is required, the analytic content being a weighted Gronwall inequality and the probabilistic content an exploration telescoping.

In the sensitive (non-saturated) regime, the paper proves two-sided finite-matrix criticality certificates: cell-partitioned inf/sup matrices $\kappa<1$02 sandwich the Perron root of the continuous-type mean operator, converge to it, and yield verifiable finite-algebra certificates of sub- or supercriticality at every non-critical parameter point. This dissolves the criticality question, though not the limit theory, of the sensitive regime into finite linear algebra.

Limitations and open problems

The manuscript is explicit about its conditionality. The reduction hypothesis is proved unconditionally only on the dissipative region $\kappa<1$03 and exactly on the Bethe lattice; the full saturated regime, including the strong-coupling case $\kappa<1$04 where the weighted-Gronwall contraction is lost and echoes can in principle resonate, remains Conjecture 9.1, which the author identifies as the substantive open problem. Uniformity through the critical window without the joint limit $\kappa<1$05 — replacing the cluster-volume factor $\kappa<1$06 by the front width $\kappa<1$07 — is plausible but not proved by the exploration telescoping. Unbounded degrees (relaxing $\kappa<1$08 to weighted-norm conditions under exponential degree tails) are open. The exponential-moment hypothesis in the CRT theorem is expected to relax to $\kappa<1$09 but is not established here. The sensitive-regime limit theory ($\kappa<1$10 with continuous types, boundary dimension, spatial CLT and Perron projection) is open, as are finite-size scaling against the quasi-stationary cutoff, non-tree unimodular limits where short cycles do not vanish, the multifractal fluctuation theory on $\kappa<1$11 (requiring essential-spectrum and Weyl-decomposition tools rather than compactness), and a travelling-wave/hydrodynamic description of the front profile itself, for which the dissipative regime with its uniform spectral gap is the natural starting point.

Conclusion

This paper reorganizes the theory of threshold cascades of interacting diffusions around a single observation — that cluster size, survival, and stability are boundary-blind — and shows that, conditional on an explicitly isolated reduction hypothesis, the entire mean-field universality picture (Perron criticality, the $\kappa<1$12 law, the CRT shape), together with the boundary geometry of the failed cluster (the pressure function, the Jensen gap, and the dimension identity $\kappa<1$13), follows from elementary finite-dimensional Perron–Frobenius theory. It simultaneously proves that the infinite-dimensional operator programme is generically impossible, that several natural identities (scalar many-to-one, spectral-radius/dimension equality, subcritical convexity, multiplicative small-noise correction) are false, and that the reduction itself is a theorem on an explicit nonempty open parameter region via a quantitative front-tracking argument. The remaining gap between the dissipative regime and the full saturated regime is precisely scoped and constitutes the paper's principal open problem.