Branching-Ruin Number & Polynomial Growth

• Branching-Ruin Number is a dimension-like invariant measuring the polynomial growth of infinite trees via optimized cutset sums.
• It sets the precise threshold for phase transitions in self-interacting processes, such as once-reinforced random walks and heavy-tailed conductance models.
• Its computation through Hausdorff dimension and generation scaling provides actionable insights into the behavior of stochastic processes on trees.

The branching-ruin number is a critical parameter that describes the polynomial growth of infinite, locally finite rooted trees and serves as the key threshold for phase transitions in various self-interacting and random processes on such trees. It was introduced to elucidate the precise criterion governing the recurrence/transience transition of the once-reinforced random walk (ORRW), and it provides a rigorous and computable analog of the classical branching number but tailored to polynomial rather than exponential growth. The concept also appears as a heavy-tailed aggregate in branching Brownian motion, where it precisely quantifies extremal “ruin” statistics.

1. Formal Definition and Motivating Framework

Given an infinite, locally finite rooted tree 𝒢 =(V,E) with root ρ, each vertex ν is assigned its graph distance from the root as |ν|, and each edge e is oriented away from ρ as e = [e⁻,e⁺] with |e⁺| = |e⁻|+1, so |e| := |e⁺|. A cutset π ⊂ E is a minimal subset of edges separating ρ from infinity, i.e., every infinite self-avoiding ray from ρ crosses exactly one edge of π. The set of all cutsets is denoted Π.

The branching-ruin number, denoted brᵣ(𝒢), is defined as

$$br_r(\mathcal G) = \sup\Bigl\{\lambda>0:\;\inf_{\pi\in\Pi} \sum_{e\in\pi}|e|^{-\lambda} > 0 \Bigr\}$$

or, equivalently,

$$br_r(\mathcal G) = \inf\Bigl\{\lambda>0: \inf_{\pi\in\Pi}\sum_{e\in\pi}|e|^{-\lambda} = 0\Bigr\}.$$

This constitutes the largest exponent λ for which the so-called ζ-series ∑ₑ∈π |e|^{−λ}, optimized across all cutsets, is strictly bounded below. The branching-ruin number thus captures polynomial growth in generations, in contrast to the branching number (which replaces |e|^{−λ} with e^{−λ|e|} and measures exponential growth) (Collevecchio et al., 2017, Collevecchio et al., 2018).

An alternative characterization is through the Hausdorff dimension of the boundary ∂𝒢 of infinite rays, equipped with the metric d(ξ,η) = 1/|e|, where e is their last common edge. Thus,

$$br_r(\mathcal G) = \dim_{\rm Hausdorff}(\partial\mathcal G,d).$$

2. Key Properties and Computation

Monotonicity holds: if T′ is a subtree of T, then brᵣ(T′) ≤ brᵣ(T).

In spherically symmetric trees, if the n-th generation Eₙ has size |Eₙ| ≈ C nᵇ, then brᵣ(𝒢)=b. In general, brᵣ(𝒢) ≤ Pgr(𝒢), where Pgr(𝒢)=lim infₙ ln|Eₙ|/ln n is the polynomial growth exponent, and there is equality for spherically symmetric cases.

Comparison with the branching number: For trees with exponential growth (br(𝒢) > 1), brᵣ(𝒢)=∞, indicating no non-trivial polynomial scaling. Conversely, if generations increase as n^b with br(𝒢)=1, then brᵣ(𝒢)=b.

Computation of brᵣ(𝒢) often proceeds by choosing a level n and estimating

$\sum_{n=1}^\infty |E_n|\, n^{-\lambda},$

which converges or diverges depending on λ in relation to the actual polynomial growth of the tree.

A summary table of growth parameters is as follows:

Parameter	Definition Logic	Captures
Branching number br(𝒢)	Replaces	e
Branching-ruin number	Uses	e
Polynomial growth exp.	liminfₙ ln	Eₙ

(Collevecchio et al., 2017, Collevecchio et al., 2018)

3. Criticality in Random Processes on Trees

Once-Reinforced Random Walk (ORRW)

For ORRW on 𝒢 with parameter δ>0 (edges have initial weights 1, change on first crossing to δ, and transitions are proportional to edge weights), the critical parameter δ_c for the recurrence/transience phase transition is exactly the branching-ruin number:

• Transient if δ < brᵣ(𝒢)
• Recurrent if δ > brᵣ(𝒢)

At criticality (δ = brᵣ), further sharp dichotomies depend on refined sum conditions on the cutsets. The probability to escape from the root to infinity without return is fundamentally dictated by the polynomial decay encoded in brᵣ(𝒢) (Collevecchio et al., 2017, Collevecchio et al., 2018).

Proof strategy: For δ>brᵣ, recurrence follows from Borel–Cantelli arguments on bounds for crossing probabilities. For δ<brᵣ, the walk is embedded into a network with modified conductances c(e) ≈ |e|^{1−δ}, and the existence of a unit flow of finite energy is established, guaranteeing transience via the max-flow/min-cut principle.

Heavy-Tailed Random Conductance Walks

Assign i.i.d. positive conductances {Cₑ} to edges, with distribution tail P(Cₑ ≤ t) ~ t^p L(t) at t→0 for some p > 1. The phase transition is controlled by the branching-ruin number: the walk is recurrent if and only if p > brᵣ(𝒢) (Collevecchio et al., 2018).

Multi-Excited (Digging) Random Walks

At each vertex place M infinite-strength cookies. Each time the walker visits a vertex with available cookies, it is forced toward the parent; otherwise, transitions are uniform. The phase transition is governed by brᵣ(𝒢) such that the critical number of cookies is M_c = brᵣ(𝒢)–1 (Collevecchio et al., 2018).

4. Explicit Examples and Case Studies

• Regular Trees of degree D > 1: Exponential growth gives brᵣ=∞; all ORRW are transient.
• Polynomial Growth Trees: If |Eₙ| ~ n^b, then brᵣ=b, and criticality in random processes matches b.
• ℤ^d–like Trees: Trees which branch d-fold at generations 2^k (otherwise singly linear) yield brᵣ = log₂d.
• Random Polynomial Trees: With offspring mean m>0, brᵣ = m almost surely for the random tree, and criticality thresholds match direct brᵣ computation.
• Comb-like Trees: The growth of teeth of length ℓₙ at generation n yields brᵣ = liminfₙ (log n)/(log ℓₙ) (Collevecchio et al., 2017, Collevecchio et al., 2018).

5. Branching-Ruin Number in Branching Brownian Motion

In the context of binary branching Brownian motion (BBM) at the boundary case (drift μ=2, diffusion σ²=2), the branching-ruin number is the total number N of birth events on the negative half-line. Formally,

$$N = \sum_{u \in \mathcal{U}} 1_{ \{ X_{d_u}(u) \leq 0 \} }$$

where 𝒰 is the Ulam–Harris tree indexing particles and X_{d_u}(u) is u's location at death (i.e., its children's birth position) (Chen et al., 2021).

The tail of N satisfies, as n→∞,

$P(N>n) = \frac{1}{n}+o\left(\frac{1}{n}\right),$

i.e., N has a Pareto tail of index 1, signifying markedly heavier tails than for classical one-particle ruin problems, where decay is typically n^{-½} or faster. The Laplace transform of N admits the precise small-λ expansion

$$-\log E[e^{-\lambda N}] = \lambda\,\log(\tfrac{1}{\lambda}) + (1 + \log 2 + \gamma)\lambda + o(\lambda)$$

where γ is Euler–Mascheroni constant (Chen et al., 2021).

This demonstrates that genealogical aggregation in branching processes amplifies the “ruin” event’s tail, yielding polynomial decay steeper than in non-branching analogs.

6. Broader Context, Extensions, and Related Invariants

The branching-ruin number connects with:

• The theory of self-interacting random walks, notably phase transition mechanisms in environments with polynomial generation growth.
• The Hausdorff dimension of the boundary in the natural metric, providing a geometric interpretation.
• Comparison to the branching number and polynomial growth exponent, giving a hierarchy of growth invariants for trees.
• Extensions to other random processes (multi-excited RW, heavy-tailed RWRC), where the critical exponent for the process matches brᵣ(𝒢) or a direct function thereof (Collevecchio et al., 2017, Collevecchio et al., 2018).
• Analogues in branching random walks and critical diffusion, where process-specific branching-ruin statistics control extremal behavior in the system (Chen et al., 2021).

In summary, the branching-ruin number brᵣ(𝒢) is a dimension-type invariant uniquely capturing the polynomial growth of trees and precisely characterizing the critical phase transition for a broad class of self-interacting and random processes on trees. Its definition via cutset sums, equivalence with Hausdorff dimension, and direct computability in broad examples underscore its foundational status in probabilistic and geometric analysis of tree-indexed stochastic processes (Collevecchio et al., 2017, Collevecchio et al., 2018, Chen et al., 2021).