Subcritical Branching Random Walk

Updated 23 August 2025

Subcritical branching random walk is a model where particles perform spatial random walks and reproduce with a mean offspring number below one, ensuring almost sure extinction yet exhibiting rich rare-event dynamics.
The framework combines random walk theory and branching processes using spectral analysis and asymptotic methods to rigorously describe decay rates and spatial profiles.
Recent studies show that incorporating heavy-tailed motions and random environments alters tail behavior and survival probabilities, offering insights into phase transitions and steady-state dynamics.

A subcritical branching random walk is a probabilistic model in which particles move in a space (typically a lattice) according to a specified random walk and branch (reproduce or die) according to a discrete–time or continuous–time branching process, with the key defining property that the expected number of offspring per particle is strictly less than one. This regime ensures that, although the total population dies out almost surely, the structure of rare events such as extreme displacement, large local populations, or the profile of surviving particles can display highly nontrivial and rich behaviors. Recent developments have elucidated the classification, sharp asymptotics, phase transitions, and deep connections with random walk theory, especially in the presence of heavy-tailed spatial steps or in random environments.

1. Definition and Classification

A branching random walk (BRW) consists of particles which evolve as follows:

Each particle, at discrete or continuous times, moves according to a specified (possibly heavy-tailed) random walk on a space such as $\mathbb{Z}^d$ .
Upon hitting special locations (or everywhere, in the classical BRW), a particle branches into a random number of offspring, distributed according to a fixed offspring law $\{p_k\}$ , or possibly a space– and/or time–dependent law.
In the subcritical regime, defined by mean offspring number $m = \sum_{k} k p_k < 1$ , the process becomes extinct almost surely.

Subcritical BRW is further differentiated by environmental or structural features:

Standard BRW: Branching occurs everywhere with a fixed $m < 1$ .
Catalytic BRW: Branching restricted to catalytic sites (e.g., the origin); reproduction laws may be subcritical even at catalysts.
Random Environment: Either the offspring law or spatial motion is modulated by a random environment, giving rise to further distinctions such as strongly, weakly, or intermediately subcritical cases, depending on large deviation and spectral properties (Afanasyev et al., 2010, Vatutin et al., 2011).

The classification becomes subtler when spatial motion is heavy-tailed or the environment is random: the definition of "criticality" requires spectral analysis of operators that encode both the branching and movement components (Rytova et al., 2020).

2. Mathematical Structure and Main Asymptotic Results

The number of particles at a given site or the maximal displacement in the subcritical BRW is governed by coupled equations for moments and probabilities, whose asymptotics depend crucially on branching and motion.

First and Higher Moments

For a spatially inhomogeneous BRW with branching localized at the origin (source), the mean number $m_1(t,x,y)$ of particles at $y$ at time $t$ , originating from $x$ , satisfies:

$\frac{d}{dt} m_1(t,x,y) = (H_B m_1(t,\cdot,y))(x),\qquad m_1(0,x,y) = \delta_y(x)$

where $H_B$ is an operator that combines the random walk (generator $A$ ) and local branching rate $B$ via $H_B = A + B \delta_0$ (Rytova et al., 2020). In the subcritical regime ( $B < B_c$ ), $m_1(t,x,y)$ decays to zero as $t \to \infty$ ; all higher moments vanish similarly.

Asymptotic rates depend on the interplay between dimension $d$ and the heavy-tail exponent $\alpha$ of the jump distribution. For a stable law with $a(z)\sim H(z/|z|)/|z|^{d+\alpha}$ :

The transition probability decays as $t^{-d/\alpha}$ .
For $B < B_c$ , $m_1(t,x,y)$ decays at a rate controlled by $d/\alpha$ ; explicit bounds are provided, e.g., $t^{1/\alpha-1}$ for certain ranges (Rytova et al., 2020).

Tail Behavior and Extremes

The probability that the maximal displacement $M_n = \sup_{k \leq n} M_k$ exceeds a threshold, or that the population survives at a distant site, exhibits exponential or subexponential (in some cases, power law) decay:

With suitable moment conditions, for $c > 0$ ,

$\lim_{n\to\infty} e^{\gamma c n} \mathbb{P}(M_n \geq c n) = \begin{cases} \kappa \in (0,1] & \text{if } c < m \mathbb{E}[X e^{\gamma X}] \[1ex] 0 & \text{if } c > m \mathbb{E}[X e^{\gamma X}] \end{cases}$

where $\gamma > 0$ is defined via $\mathbb{E}[e^{\gamma X}] = 1/m$ and $\kappa$ is a nontrivial constant (Hou et al., 21 Aug 2025).

For catalytic BRWs with simple symmetric random walk and $m < 1$ , the tail is heavy:

$(1-a) P_0(M > x) \sim \frac{2a(1-m)}{x}, \quad x \to \infty$

demonstrating the polynomial decay of the maximal displacement (Bulinskaya, 2020).

For killed BRWs (where particles are removed upon reaching a boundary), the total progeny $Z$ satisfies:

$\mathbb{P}_x(Z > n) \sim c_{\mathrm{sub}} R(x) e^{\varrho_+ x} n^{-\varrho_+/\varrho_-}, \quad n \to \infty$

where $R(\cdot)$ is a renewal function and $\varrho_-, \varrho_+$ are roots of the associated Cramér function (Aïdékon et al., 2011).

3. Effects of Heavy-Tailed Motion and Spatial Mechanisms

A defining feature in many subcritical BRW studies is the inclusion of heavy-tailed or long-range jumps. Transition intensities with power–law decay ( $|y-x|^{-d-\alpha}$ , $\alpha\in(0,2)$ ) render the underlying random walk of infinite variance, altering fundamental behaviors:

Finite-variance cases only admit a nontrivial critical branching threshold ( $B_c>0$ ) for $d\geq3$ , whereas in the heavy-tailed case, $B_c$ can be positive for much lower $d$ , specifically when $d/\alpha > 1$ (Rytova et al., 2020).
Long-range jumps slow spatial decay (from $t^{-d/2}$ to $t^{-d/\alpha}$ ), influence the spectrum of the evolution operator, and thus shift the critical/subcritical dichotomy.
For catalytic models, the structural constraint of branching at rare sites (e.g., only at the origin) changes tail phenomena from exponential to polynomial decay—even in the subcritical regime—showing qualitatively new effects (Bulinskaya, 2020).

4. Random Environment and Conditional Behaviors

Random or time-varying environments can dramatically affect subcritical BRW properties:

In branching processes in random environments (BPREs), the notion of (weak, strong, intermediate) subcriticality is formalized via the associated random walk $S_n = \sum_{k=1}^n \log m(Q_k)$ , with $Q_k$ i.i.d. environment variables (Afanasyev et al., 2010).
Asymptotic survival relations tie the survival probability to the random walk staying above zero:

$\mathbb{P}\{Z_n > 0\} \sim K\, \mathbb{P}\left\{\min_{1\leq k \leq n} S_k \geq 0\right\}$

In the weakly subcritical case, conditioned on non-extinction, the process demonstrates "conditional supercriticality": surviving lines behave like a typical supercritical process for intermediate times, a phenomenon made explicit by functional limit theorems (Afanasyev et al., 2010).
Absence of the Cramér condition (i.e., heavy-tailed environment) modifies survival entirely: persistence for large $n$ is essentially due to a single atypical "big jump" in the random walk, after which the system decays along a deterministic trajectory with Brownian fluctuations on the logarithmic scale (Vatutin et al., 2011).

5. Extremes, Martingales, and Rare Events

Analysis of extremes and associated martingales is central for understanding rare events and the tail behavior of subcritical BRW:

The minimal and maximal positions relate to derivative and additive martingales, whose behaviors differ sharply in tail decay and localization.
In Gaussian branching random walks, precise estimates for the left tail of the derivative martingale establish stretched exponential decay with exponent $\gamma = (\beta_c/\beta)^2$ , confirming deep conjectures and evincing the thinness of the left tail compared to the heavy right tail (Bonnefont et al., 2023).
In killed or conditioned BRWs, renewal and change-of-measure techniques akin to those developed by Aïdékon, Hu, and Zindy are key to deriving such extreme value asymptotics (Aïdékon et al., 2011, Hou et al., 21 Aug 2025).

6. Spatial Profile, Empty Regions, and Statistical Equilibrium

Spatial aspects of subcritical BRW include:

The radius $R_n$ of the largest empty ball (no particles inside) admits explicit scaling limits that depend on both the offspring law and dimension. In the subcritical regime, $R_n$ grows exponentially with $n$ , and its properly scaled distribution converges to an explicitly characterized law expressing the exponential rarity of distant particles (Xiong et al., 2022).
For contact branching random walks with arbitrary offspring distribution and immigration, subcriticality (mortality exceeding branching rate and tail decay of offspring law) ensures convergence to a unique spatial equilibrium (steady–state) particle field (Chernousova et al., 2018).
Dimensionality and offspring variance (finite/infinite) critically determine the decay rates in local occupation probabilities and moments, often leading to power–law or logarithmic corrections in low dimensions (Bulinskaya, 2012, Rytova et al., 2020).

7. Duality, Extensions, and Future Directions

A celebrated duality links subcritical maximal displacement to that in the supercritical regime conditioned on extinction: the main tail estimates for subcritical BRW carry over to the supercritical case conditioned on extinction, under appropriate transformation of the offspring law (Neuman et al., 2015).
Extensions to multitype processes, non-lattice spatial domains, general (possibly Lévy) motion, and non-i.i.d. random environments broaden the landscape of subcritical BRW paper.
Statistical steady states and spatial intermittency in models with immigration, as well as the impact of environmental or spatial randomness, are fertile areas for continued research (Chernousova et al., 2018, Vatutin et al., 2017, Hong et al., 2018).

These structural and asymptotic results define the core of the modern theory of subcritical branching random walks, encompassing the critical influence of heavy-tailed motion, the subtleties of random environment, and the multifaceted behaviors revealed by rare events, spatial extremes, and equilibrium statistical properties. Analytical tools—ranging from operator spectral theory, martingale methods, and Tauberian theorems to renewal and change-of-measure arguments—enable a fully quantitative understanding of both typical decay and exceptional events for subcritical systems.