Branching Random Walks (BRW)

Updated 17 April 2026

Branching random walks (BRW) are stochastic processes that combine random motion with reproduction, capturing both spatial dispersion and genealogical structures.
BRWs analyze survival regimes, including global and local survival, with phase transitions influenced by branching rates and the geometry of the underlying space.
BRWs exhibit complex fractal and multifractal behavior in their spatial traces, providing insights into propagation thresholds and extreme value phenomena.

A branching random walk (BRW) is a stochastic process in which particles proliferate and disperse across a state space according to prescribed probabilistic laws. The process combines both a spatial motion (random walk) and reproduction (branching), resulting in a multi-type Markov process indexed by a tree of genealogies and locations. BRWs generalize classical branching processes by incorporating the spatial structure of random walks, which allows for the analysis of spread, recurrence, survival phases, and fractal geometry of the set of visited sites. These properties have critical implications in mathematics and applications ranging from epidemiology to statistical physics.

1. Mathematical Definition and Structure

A typical BRW is defined over a (possibly infinite) connected graph or a group, such as ℤ^d or a Cayley graph. The process evolves in discrete or continuous time. At the n-th time step or generation, the system is described by the population configuration—each particle is characterized by its spatial position and genealogy.

Formally, for a space $X$ , a BRW can be specified as follows (Bertacchi et al., 2011, Bertacchi et al., 2015):

Each particle at site $x$ produces offspring according to a site-dependent (or site-independent) law $\mu_x$ : a probability measure on finite-support functions $f: X \to \mathbb{N}$ with $\sum_{y} f(y) < \infty$ .
For the continuous-time BRW, each particle may die at rate 1 and have children at rates specified by a kernel $k_{xy}$ .
The process is governed by its first-moment matrix $M = (m_{xy})$ where $m_{xy}$ is the expected number of offspring a particle at $x$ sends to $y$ .

A multidimensional generating function $x$ 0 captures the full dynamics: $x$ 1 Extinction probabilities and other statistics are characterized as fixed points of $x$ 2 (Bertacchi et al., 2015). For spatially homogeneous BRWs, the configuration reduces to a time-indexed point process on $x$ 3 or a discrete graph.

2. Survival, Recurrence, and Phase Transitions

A central theme in BRW theory is the classification of regimes by survival properties, dictated by branching intensities, offspring distributions, and the geometry of the underlying space (Bertacchi et al., 2011, Bertacchi et al., 2011, Ajax et al., 9 Dec 2025).

Types of survival behavior:

Global survival: The probability that the total population avoids extinction.
Local survival: The event that a particle visits (or returns to) a specific site infinitely often.
Strong local survival: Conditioned on global survival, the process continues to visit particular locations infinitely often.

For continuous-time BRWs on graphs, two critical parameters are introduced: $x$ 4

$x$ 5

Pure global survival means $x$ 6; the process can survive globally while any fixed vertex is eventually vacated (Bertacchi et al., 2011, Bertacchi et al., 2012). In irreducible, quasi-transitive cases, local survival implies strong local survival, but strong local survival may fail to be monotone in parameters such as branching rate (Bertacchi et al., 2012).

Phase transitions in these critical parameters reflect shifts between extinction, strong local survival, and non-strong ("pure global") survival phases. For example, BRWs on non-amenable graphs (e.g., trees or free products of groups) can show these nontrivial regimes; local modifications (e.g., adding fast loops at a site) can change the phase diagram (Bertacchi et al., 2015).

3. Extremal Behavior, Maxima, and Fluctuations

For BRWs with real-valued increments (typically on trees), precise asymptotics of extremal positions reveal universality and deep connections with random energy models and extreme-value theory (Kistler et al., 2015, Bandyopadhyay et al., 2021, Kowalski, 2023).

On regular binary trees with Gaussian increments (the canonical BRW), the maximum position $x$ 7 at generation $x$ 8 satisfies

$x$ 9

As $\mu_x$ 0, the extremal process converges to a decorated Poisson cluster process driven by a derivative martingale (Kistler et al., 2015).

Models with time or genealogy inhomogeneity, non-Gaussian increments, or end-point perturbations exhibit shifted maxima and, in some regimes, logarithmic corrections ( $\mu_x$ 1 with $\mu_x$ 2 determined by the regime and cumulant) (Bandyopadhyay et al., 2021, Kowalski, 2023, Ouimet, 2015). For example, "last progeny modified BRWs" or perturbated models maintain the same LLN for the maxima but adjust the logarithmic correction and limit law according to explicit analytical martingale computations.

4. Geometric and Fractal Properties of the Trace

The trace of a BRW, i.e., the random subgraph of all sites and edges ever visited, exhibits complex fractal structure, especially on non-amenable and hyperbolic spaces (Candellero et al., 2011, Lai et al., 2024, Duquesne et al., 2021).

Key phenomena:

On Cayley graphs of free products, the trace in the transient regime ( $\mu_x$ 3) is a proper subgraph. Its end-boundary set $\mu_x$ 4 is a random fractal subset of the boundary space $\mu_x$ 5.
The box-counting and Hausdorff dimension of the limit set $\mu_x$ 6 coincide and can be computed via an explicit generating function formula; the dimension $\mu_x$ 7 varies continuously in $\mu_x$ 8 except for a single discontinuity at the critical point $\mu_x$ 9, where the trace fills the whole graph (Candellero et al., 2011).
When branching occurs on free groups, the limit set allows for a multifractal decomposition: For each escape rate $f: X \to \mathbb{N}$ 0, the Hausdorff dimension of the set $f: X \to \mathbb{N}$ 1 is given by explicit LV-type Legendre transforms, yielding a multifractal spectrum (Lai et al., 2024).
For BRWs indexed by critical Galton–Watson trees, the range scaled appropriately converges (in the sense of metric measure spaces) to continuum random trees or the Brownian cactus, reflecting the interplay of genealogy and spatial motion (Duquesne et al., 2021).

5. Large Deviations, Multifractality, and Free Energy

A multifractal theory emerges from the description of level sets of BRWs, particularly with real or vector-valued increments in continuous or discrete spaces (Attia et al., 2013, Lai et al., 2024). This analysis centers on the empirical averages along paths and the dimensions of the sets of rays for which these averages have specified limiting points.

For a supercritical BRW with increments in $f: X \to \mathbb{N}$ 2, the Hausdorff and packing dimensions of the sets of rays for which the empirical mean converges to a set $f: X \to \mathbb{N}$ 3 are given by the Legendre–Fenchel transform of the logarithmic moment generating function (the "free energy") (Attia et al., 2013).
Phase transitions emerge in the multifractal spectrum, corresponding to first- or second-order singularities in the free-energy function, resulting in plateaus or kinks in singularity spectra (Attia et al., 2013).
On non-amenable graphs (e.g., free groups), the rate function and pressure enter the dimension formula for the boundary limit sets, and these can be computed via large-deviation techniques and thermodynamic formalism (Lai et al., 2024).

This multifractal framework extends to Mandelbrot measures and the $f: X \to \mathbb{N}$ 4-spectrum, yielding a comprehensive description of the measure-theoretic complexity generated by the BRW.

6. Applications, Generalizations, and Open Problems

BRWs provide key models for the spatial spread of epidemics, population genetics, and information in networks. For instance, continuous-time BRWs on $f: X \to \mathbb{N}$ 5 with catalytic (localized) branching have been proposed and rigorously analyzed for propagation and phase thresholds relevant to epidemiology (Ermakova et al., 2019, Yarovaya, 2017). The spectral dimension and geometry of the underlying space or network critically impact the growth rate, volume explored, and scalability of outbreaks (Bordeu et al., 2019).

Recent work has extended BRWs to allow inhomogeneities in both time and genealogy, leading to non-trivial spatial structures and connections with pattern formation in biology (e.g., bacterial colonies), with the suggestion that new traveling wave and fractal regimes arise from such models (Ajax et al., 9 Dec 2025).

Key open problems include:

Extension of the explicit trace/fractal geometry results from free products to broader classes of non-amenable or hyperbolic groups (Candellero et al., 2011).
Rigorous analysis of the multifractal spectrum for the full limit measures (not only the sets) on the boundary (Lai et al., 2024).
Development of robust large-deviation and scaling methods for inhomogeneous, genealogy-driven, or density-dependent BRWs (Ajax et al., 9 Dec 2025).
Analysis of interacting particle systems related to BRWs, such as contact and Ising processes, with respect to analogous phase transitions (Candellero et al., 2011).