Poisson(1) Branching Process: A Critical Model

• The Poisson(1) branching process is a critical single-type Galton–Watson model characterized by a Poisson offspring distribution that almost surely leads to extinction.
• It features a fixed-point equation q = e^(q-1) and a series representation that connects classical Cayley tree enumeration with random combinatorial structures.
• The total progeny follows the Borel distribution with heavy-tailed behavior, offering insights into random graphs, critical phenomena, and vector coalescence frameworks.

A Poisson(1) branching process is the canonical example of a single-type Galton–Watson process with a critical Poisson offspring distribution where each individual has a number of children distributed as $\operatorname{Poisson}(1)$. This process serves as a paradigmatic case of critical branching exhibiting almost sure extinction but with infinite expected total progeny, and features combinatorial connections to classical enumeration of trees and the Borel distribution. All classical properties of the one-type Poisson(1) Galton–Watson process can be derived as specializations of the multi-type Poisson branching–process framework and its correspondence to vector multiplicative coalescent equations, as elaborated in the work of Aravinda, Kovchegov, Otto, and Sarkar (Aravinda et al., 2024).

1. Reduction from Multi-Type Branching to the Single-Type Fixed Point Equation

In the multi-type ($k$-type) Poisson branching process with mean matrix $\mathbf{M} = (m_{ij})$, the offspring generating function for type $i$ is

$$f_i(s_1,\dots,s_k) = \exp \left( \sum_{j=1}^k m_{ij}(s_j - 1) \right).$$

The extinction vector $\boldsymbol{\eta} = (\eta_1,\dots,\eta_k)$ is the minimal solution in $[0,1]^k$ to the system: $$f_i(\boldsymbol{\eta}) = \eta_i,\quad i = 1,\dots,k.$$ Specializing to the single-type case ($k=1$), where $m_{11} = m$, this reduces to the scalar fixed-point equation

$\eta = \exp(m(\eta-1)) =: G(\eta).$

Setting $m = \lambda$ (the Poisson$(\lambda)$ case), the extinction probability $q$ solves

$q = e^{\lambda(q-1)}.$

The Poisson(1) process corresponds to the critical point $\lambda = 1$: $\boxed{q = e^{q-1}}, \qquad 0 \leq q \leq 1.$ A direct verification shows the unique solution is $q = 1$, and hence extinction occurs almost surely (Aravinda et al., 2024).

2. Series Representation for the Extinction Probability

While the critical case admits the explicit value $q=1$, the series expansion derived by Aravinda–Kovchegov–Otto–Sarkar is instructive and connects to classical combinatorics. Specializing their general formula (see Theorem 4.2 in (Aravinda et al., 2024)) to $k=1$, $\alpha=1$, $t=1$ gives: $q = \sum_{x=1}^\infty \frac{x^{x-1} e^{-x}}{x!}.$ This series enumerates rooted labeled trees ("Cayley trees") with the factor $x^{x-1}/x!$. The first few terms are

$$q = e^{-1} + e^{-2} + \frac{9}{6} e^{-3} + \frac{64}{24} e^{-4} + \cdots,$$

with partial sums converging to 1. This convergence, with explicitly combinatorial coefficients, highlights the connection to enumeration of trees in random graphs and stochastic processes (Aravinda et al., 2024).

3. Total Progeny: Generating Function and Borel Law

Let $T$ denote the total progeny (including the original ancestor). Its generating function $F(s)$ solves: $F(s) = s \exp(F(s) - 1)$ which is equivalent to $F(s) e^{1 - F(s)} = s$. Employing Lagrange inversion or the series for the principal branch of the Lambert $W$-function, the explicit closed form is: $F(s) = -W(-s e^{-1}),$ where $W$ is the principal value of the Lambert $W$-function. The power series expansion yields

$$F(s) = \sum_{n=1}^\infty \left[ e^{-n} \frac{n^{n-1}}{n!} \right] s^n,$$

and thus for $n \geq 1$,

$\boxed{\Pr(T=n) = e^{-n} \frac{n^{n-1}}{n!}}$

This is the classical Borel distribution (parameter 1) for total progeny in the critical Poisson(1) Galton–Watson process (Aravinda et al., 2024).

4. Critical Phenomena at $\lambda=1$

The process at the critical point $\lambda=1$ exhibits the following features:

• The extinction probability is $q=1$, so the process dies out with probability one.
• The mean total progeny diverges: $$\mathbb{E}[T] = \sum_{n=1}^\infty n\,\Pr(T=n) = \sum_{n=1}^\infty e^{-n} \frac{n^n}{n!} = +\infty$$ due to the Borel tail decaying as $n^{-3/2}$.
• The singularity structure of $F(s)$ around $s=1$ is of square-root type: $$F(s) = 1 - \sqrt{2(1-s)} + O(1-s), \quad s \rightarrow 1^-,$$ implying

$\Pr(T=n) \sim C n^{-3/2}$

for some constant $C$. This heavy-tailed behavior is characteristic of critical Galton–Watson processes and underlies critical behavior in random combinatorial structures (Aravinda et al., 2024).

5. Combinatorial Context and Connections

The coefficients $n^{n-1}/n!$ in both the extinction probability series and the progeny law correspond to the enumeration of rooted labeled trees, connecting the Poisson(1) branching process to the classical Cayley formula. This suggests deep relations with random graph theory, notably in the enumeration of components and trees in Erdős–Rényi graphs at the critical window, and highlights the broader applications of branching process tools in random combinatorial structures (Aravinda et al., 2024).

6. Vector Coalescence Correspondence

The extinction equations for the Poisson($\lambda$) branching process are equivalent to the gelation equations for the vector (multi-type) multiplicative coalescence process. The vector coalescent and the single-type branching process share a fixed-point system, with gelation in the coalescent corresponding to almost sure extinction in the branching model. This correspondence allows cross-applications: for example, the extinction series for the branching process provides a new series solution for the gelation problem, as established in Aravinda–Kovchegov–Otto–Sarkar (Aravinda et al., 2024).

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All detailed results, intermediate equations, and combinatorial interpretations summarized here follow directly from (Aravinda et al., 2024).