Critical Branching Random Walk

• Critical branching random walk is a particle system where each particle reproduces with a mean one offspring and moves independently, ensuring a delicate balance between proliferation and extinction.
• It underpins studies in spatial population dynamics, epidemic thresholds, and random structure geometry through lattice, graph, and continuous-time models.
• Research on BRWs focuses on phase transitions, range scaling, branching capacity, and extremal displacement, offering insights applicable in probability, physics, and theoretical computer science.

A critical branching random walk (BRW) is a discrete- or continuous-time particle system where individual particles reproduce according to a critical branching process and move independently according to a Markovian random walk, typically on a lattice or a graph. The system is said to be critical when the mean number of offspring per particle is exactly one, so that population size neither explodes nor dies out exponentially quickly, but rather exhibits intricate random fluctuations with extinction occurring almost surely in homogeneous environments. Critical BRWs are fundamental in probability, statistical physics, and theoretical computer science, underpinning phenomena in spatial population dynamics, epidemic spread at threshold, and the geometry of high-dimensional random structures.

1. Foundational Structure and Examples

A critical BRW combines two stochastic mechanisms:

• Branching Law: Each particle at each step reproduces independently, spawning a random number of offspring with mean one; prototypically a critical Galton–Watson process.
• Spatial Motion: Each offspring’s displacement is independent, sampled from a fixed jump kernel or random walk transition.

Canonical examples include:

• Lattice-based models: Particles move on $\mathbb{Z}^d$ with symmetric simple random walk increments and critical branching (Bai et al., 21 Nov 2025, Zhu, 2016, Lalley et al., 2012).
• General graphs: The underlying space $G=(V,E)$ may be an arbitrary finite- or infinite-degree graph, with a criticality criterion linked to the spectral radius of the walk’s transition kernel $P$ (Candellero et al., 2014).
• Continuous-time formulations: Branching and movement can be modeled via continuous-time Markov processes, with the total branching rate or “source intensity” tuned so that the expected number of descendants remains constant over time (Rytova et al., 2018, Yarovaya, 2017).

Such models may incorporate heavy-tailed displacement distributions (Mallein, 2015), random environments (Garet et al., 2012, Nakashima, 2012), or constraints such as population size control (“selection”) (Mallein, 2015).

2. Phase Transition, Criticality, and Survival

Critical BRWs demarcate a sharp phase transition between exponential extinction (subcritical, mean offspring $<1$) and explosive growth (supercritical, mean $>1$). In homogeneous environments, extinction is almost sure: in discrete time, this follows from the classical critical Galton–Watson theorem; for BRWs, spatial motion modulates recurrence/transience and the time scale for extinction.

• Survival probability: In one spatial dimension, for recurrent underlying walks, the survival probability decays as a power law—$Q(t)\sim C t^{-1/2}$ for finite-variance jumps, or more generally $Q(t)\sim const \cdot t^{-1/\alpha}$ for heavy-tailed walks with exponent $\alpha \in (1,2)$ (Rytova et al., 2018).
• Random environment: In space-time random environments, the critical BRW dies out almost surely when the associated free energy vanishes ($\Psi=0$), completing the trichotomy: extinction for $\Psi \leq 0$, positive probability survival for $\Psi > 0$ (Garet et al., 2012).

3. Range, Volume, and Spatial Structure

A central object of paper is the range or volume explored—the number of distinct sites visited by the BRW up to time $n$ or $t$. This quantity encapsulates both the tree and spatial geometry and displays dimension-dependent scaling (Bordeu et al., 2019, Bai et al., 21 Nov 2025, Duquesne et al., 2021).

• Law of large numbers thresholds: For BRW on $\mathbb{Z}^d$, $d=5$ is the threshold for the range to grow linearly in $n$ (Bai et al., 21 Nov 2025). For $d<5$, range and volume saturate due to recurrence; for $d\geq 5$, the mean explored volume grows linearly.
• Explored volume scaling:
  • Lattice dimension $d$: $V(t) \sim t^{(d-2)/2}$ for $2$V(t) \sim t/\log t$ for $d=4$; $V(t) \sim t$ for $d>4$ (Bordeu et al., 2019).
  • On general graphs, scaling is controlled by the spectral dimension $d_s$: $V(t) \sim t^{(d_s-2)/2}$ for $2<d_s<4$, $V(t) \sim t$ for $d_s > 4$.
  • Random trees: For critical Galton–Watson trees, $d_s=4/3$, so the expected range saturates but higher moments grow algebraically (Bordeu et al., 2019).
• Central limit theorem and variance: For geometric offspring BRW indexed by the Kesten tree on $\mathbb{Z}^d$, the range exhibits linear variance for $d>8$ and satisfies a central limit theorem for $d>16$ (Bai et al., 21 Nov 2025).

Dimension/Graph	Asymptotic range/volume scaling
$\mathbb{Z}^d$, $d<5$	Sublinear/Saturates
$\mathbb{Z}^d$, $d=5$	Linear law of large numbers
$\mathbb{Z}^d$, $d=8$	Linear variance threshold
$\mathbb{Z}^d$, $d=16$	CLT for range
General graph, $d_s>4$	Linear
Critical random tree	Saturates (mean), nontrivial moments

4. Branching Capacity, Potential Theory, and Hitting Probabilities

The extension of discrete potential theory to critical BRWs leads to the notion of branching capacity, which determines the asymptotics of rare event probabilities such as visiting a distant set (Zhu, 2016, Zhu, 2017).

• Branching capacity $\operatorname{Cap}_B(A)$: Generalizes classical capacity to accommodate tree-structured exploration. The BRW visiting probability of a finite set $A$ decays as

$$\Pr(\text{BRW from } x \text{ visits } A) \sim a_d \frac{\operatorname{Cap}_B(A)}{|x|^{d-2}}, \quad |x| \rightarrow \infty,\ d\geq5$$

with $a_d$ depending on the random walk kernel.

• Behaviour at critical dimension $d=4$: Universal logarithmic corrections appear, e.g., $P_x(\text{visits } K) \sim C/(|x|^2 \log|x|)$ (Zhu, 2017).
• Hitting distributions: Upon conditioning that the set is visited, the first hitting distribution converges to a limit law determined by “escape masses” or harmonic measure–type quantities. This limit is sensitive to the potential kernel and walk structure (Zhu, 2016, Zhu, 2017).

5. Extremal Displacement and Selection

The position of the rightmost particle (“maximal displacement”) and its scaling are central objects in BRW theory.

• Critical case $\mathbb{Z}$, driftless, finite variance: Probability of maximal displacement exceeding $x$ decays as $P\{M \geq x\} \sim 6\eta^{2}/(\sigma^2 x^2)$, with $\eta^2$ the step variance and $\sigma^2$ the offspring variance (Lalley et al., 2012).
• Branching-selection models: BRW with selection of the $N$ rightmost survives exhibits novel scaling of the front. For exponential tails, $U_N \sim v_\infty - C/(\log N)^2$; under heavy tails where the spine is $\alpha$-stable, $U_N \sim -C^* L^*(\log N) / (\log N)^\alpha$ as $N \rightarrow \infty$ (Mallein, 2015).
• Selection at critical rate: For population control $N_n \sim \exp(a n^{1/3})$, the extremal positions at time $n$ scale as $n^{1/3}$ with explicit constants depending on $a$ and the underlying kernel (Mallein, 2015).

Model	Front/Maximal displacement scaling
BRW, driftless $\mathbb{Z}$	$P\{M \geq x\} \sim x^{-2}$
Selection $N$-BRW, exponential tail	$(\log N)^{-2}$ corrections
Selection $N$-BRW, $\alpha$-stable	$(\log N)^{-\alpha}$ corrections
Critical selection, $N_n \sim \exp(a n^{1/3})$	$n^{1/3}$-scale, explicit constants

6. Geometry of the Trace and Ends

The trace of a critical BRW, as a random subgraph, is characterized by its number of topological ends—a property deeply influenced by the interplay of the underlying graph and the symmetry of the walk.

• Infinitely many ends: For quasi-symmetric walks with certain summability, the trace almost surely has infinitely many ends (Candellero et al., 2014).
• One-ended examples: Breaking symmetry (via, e.g., bias or structure of $G$) can produce one-ended traces, even at criticality.

This structure is intricately connected to the spectral radius of the random walk kernel and the recurrence properties of the underlying graph.

7. Scaling Limits, Functional Limit Theorems, and Continuum Models

Critical BRWs admit scaling limits that describe the genealogical and spatial structure in terms of continuum random trees, Brownian motion, and superprocesses.

• Scaling of tree-valued BRW: The properly rescaled range of critical BRW indexed by Galton–Watson trees converges to the Brownian cactus or its stable analog—a random real tree metric space encoding both spatial and genealogical information (Duquesne et al., 2021).
• Functional CLTs: For high dimensions, the range of the BRW satisfies a central limit theorem, converging after centering and scaling to a Gaussian distribution (Bai et al., 21 Nov 2025).
• Super-Brownian motion in random environment: Branching random walks in space-time random environments converge, under proper scaling, to super-Brownian motion with a random noise term (Nakashima, 2012).

These scaling limits link discrete critical BRWs to fundamental objects in continuum probability and statistical physics.

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Key references:

• (Bai et al., 21 Nov 2025, Zhu, 2016, Lalley et al., 2012, Candellero et al., 2014, Zhu, 2017, Mallein, 2015, Rytova et al., 2018, Garet et al., 2012, Yarovaya, 2017, Bordeu et al., 2019, Duquesne et al., 2021, Nakashima, 2012, Mallein, 2015)