Branching Random Walks

• Branching random walks are stochastic processes that combine random reproduction through a Galton–Watson framework with spatial movement via independent random steps.
• They exhibit distinct survival modes and phase transitions, with local and global survival regimes determined by spectral properties and displacement laws.
• Analytical methods such as generating function formalism and extreme-value theory provide insights into extinction probabilities and spatial growth patterns.

A branching random walk (BRW) is a probabilistic process evolving on a discrete or continuous state space where particles reproduce (branch) and displace (walk) over time. The BRW incorporates both a genealogical structure—random branching governed by an offspring law, often on a Galton–Watson tree—and spatial dynamics, where each particle performs a random walk, either in discrete or continuous time, typically with independent increments. BRWs serve as foundational models in probability, statistical mechanics, biology, and the theory of random media, capturing the interplay between population growth and spatial spread.

1. General Definitions and Model Formulations

A canonical discrete-time BRW is specified by:

• An offspring distribution $\mu$ (birth law) with $\mu(k)$ the probability a particle has $k$ children.
• A displacement law $\nu$ dictating the random walk step of each progeny.
• A corresponding first-moment matrix $M=(m_{xy})_{x,y}$, where $m_{xy}$ is the expected number of children sent from $x$ to $y$.

The process is typically constructed on a random tree $T$:

• Each node $u$ has an independent random number of children $Z_u\sim\mu$.
• Independent spatial displacements $X_v\sim\nu$ are attached to each edge.
• The position of a node $u$ is $S_u=\sum_{\varnothing \prec v \preceq u} X_v$ with root $S_\varnothing=0$.

In continuous time, a particle at $x$ lives an exponential time, sends children to $y$ at rate $\lambda k_{xy}$, and then dies. The collection of all occupied sites at time $t$ forms the spatial projection of the evolving branching tree. BRWs have been developed and analyzed on Euclidean lattices $\mathbb{Z}^d$, Cayley graphs, and general countable or even random (e.g., Galton–Watson) graphs (Bertacchi et al., 2011, Bertacchi et al., 2015, Rytova et al., 2018).

2. Survival Modes and Phase Transitions

BRWs admit three central notions of survival:

• Global survival: the population survives somewhere at all times.
• Local survival: a fixed site (or set) is visited infinitely often.
• Strong local survival: conditional on global survival, a fixed site is hit infinitely often with probability one.

The distinction between these regimes reflects intricate dependence on the spatial structure and branching dynamics. Local survival is governed by the spectral radius $\rho(M)$ of the first-moment matrix: local survival at $x$ occurs iff $\rho(M)>1$. Global survival is characterized by the existence of a nontrivial solution to $G(z)\le z$ for the generating function $G$, and is not determined solely by $M$ except in special (e.g., continuous-time or highly symmetric) cases (Zucca, 2010, Bertacchi et al., 2011, Bertacchi et al., 2015).

A remarkable phenomenon is the pure global survival phase: in nonamenable, often tree-like geometries, a regime exists where global survival occurs with positive probability even though local survival at any finite set occurs with zero probability. The width of the pure global phase (i.e., the interval in $\lambda$ where global survival occurs without local survival) is determined by the gap between $\lambda_w$ (global) and $\lambda_s$ (local) critical thresholds (Bertacchi et al., 19 Jul 2025, Su, 2013).

3. Generating Function Formalism and Fixed-Point Structure

The dynamics and extinction behavior of BRWs are encoded in an infinite-dimensional generating function: $$G_x(\mathbf{s}) = \sum_{f\in S_X} \mu_x(f) \prod_{y\in X} \mathbf{s}(y)^{f(y)}$$ with $G: [0,1]^X \to [0,1]^X$, typically continuous and coordinate-wise monotone. Fixed points of $G$ play a central role:

• The global extinction vector $\overline{\mathbf{q}} = \mathbf{q}(\cdot,X)$ is the smallest fixed point.
• For each subset $A\subseteq X$, extinction in $A$ corresponds to a fixed point $\mathbf{q}(\cdot, A)$.

In one-dimensional (Galton–Watson) branching, $G$ is convex and has at most two fixed points. In the multi-type or infinite setting, $G$ can have uncountably many fixed points, many of which are not extinction probabilities for any subset. The lack of convexity and the proliferation of fixed points are generic for infinite-type BRWs (Bertacchi et al., 2015, Bertacchi et al., 2018).

A significant result is the construction of irreducible BRWs with uncountably many distinct extinction probability vectors—an impossibility in the single-site or finite-state setting. For example, on the regular tree $\mathbb{T}_d$, the edge-breeding BRW in the pure global regime realizes an uncountable family of extinction vectors, each corresponding to extinction in a disjoint collection of subtrees along a ray (Bertacchi et al., 2018).

4. Effects of Local Modifications and Critical Parameters

Local (finite-set) modifications of the branching or transition rates significantly affect survival thresholds and the geometry of extinction sets. Two BRWs are equivalent if their breeding matrices coincide outside a finite subset. Within an equivalence class:

• The global survival critical value $\lambda_w$ is always maximized by those BRWs that admit a pure global phase.
• Any local increase in the breeding rate can force $\lambda_s$ to coincide with $\lambda_w$, collapsing the pure global window (Bertacchi et al., 19 Jul 2025, Bertacchi et al., 2015).
• Survival or extinction probabilities in finite sets are stable under local modifications except at criticality.

These principles allow explicit manipulation of the survival diagram, as exemplified by adding strong self-loops to trees, which can transition a model from pure global to strong local survival (Bertacchi et al., 2015, Su, 2013).

5. Span, Extremes, and Geometry Under Conditioning

Conditioned BRWs and their spatial statistics reveal new universality classes. Condition on the rare event $\{Z_n=k\}$ (exactly $k$ particles in generation $n$), and analyze the limiting spatial profile as $n\to\infty$. Main findings (Bai et al., 2021):

• The law of the cut tree conditioned on $\{Z_n=k\}$ converges to a universal "small-tree" law $\st_k$ on finite trees with $k$ leaves and at least two children at the root.
• The span and gap statistics of particle positions in the terminal generation exhibit a dichotomy:
  • Critical case ($m=1$): polynomial tails for the span and gaps ($x^{-2}$ decay).
  • Noncritical ($m\ne1$): exponential tails with explicit constants dependent on the Legendre transform of the displacement and offspring law.
• These results generalize previous continuous-time findings for branching Brownian motion and demonstrate new dependencies on the full offspring distribution: for non-geometric $\mu$, gap constants depend both on $k$ and the index.

The proof strategy combines pruning/cutting procedures, ratio-limit theorems, spine decompositions, and extreme-value theory, leading to precise weak convergence results under rare-survival conditioning (Bai et al., 2021).

6. Geometry of Survival, Cover Times, and Trace

The interplay between branching structure and spatial geometry yields diverse phenomena:

• On the hypercube and expanders, the cover time of branching random walks is governed both by expansion properties and the mutation (walk) kernel. Increasing the mutation rate enhances cover time up to a critical threshold, beyond which no further speedup occurs—a saturation phenomenon (Balelli et al., 2016).
• The trace of a transient BRW on a Cayley (or unimodular random) graph is almost surely an infinite, nonamenable, exponentially growing, unimodular random subgraph which remains transient for the simple random walk and strongly recurrent for any supercritical BRW on it. The percolation critical probability $p_c$ is strictly below one (Benjamini et al., 2010).
• In high dimensions, intersection probabilities and ranges of BRWs are equivalent to Minkowski sums of independent random walks. This supports a capacity-theoretic characterization of the "branching capacity" and links the study of BRWs to deep results in potential theory and subadditive ergodic theory (Asselah et al., 2023, Asselah et al., 2023).

7. Open Problems and Directions

Current research avenues include:

• Extension of rare-survival statistics to multi-type, heavy-tailed, or non-reversible displacement regimes (Bai et al., 2021).
• Full characterization of extinction vector geometry and its connection to the structure of non-convex subsolution sets of the generating function (Bertacchi et al., 2015, Bertacchi et al., 2018).
• Elucidation of phase transitions, localization, and intermittency in random or dynamically evolving environments, with connections to the parabolic Anderson model and superprocesses (König, 2020, Nakashima, 2013).
• Understanding survival landscapes under local modifications and relating capacity-based invariants to other geometric and probabilistic features (Bertacchi et al., 19 Jul 2025, Asselah et al., 2023).

Branching random walks thus unify themes of spatial branching processes, extremal distributions, geometric group theory, and mathematical physics, with a rich structure expressed through spectral theory, generating function fixed points, and potential-theoretic concepts. Continued advances exploit both probabilistic and combinatorial frameworks, with applications extending to random media, biology, network dynamics, and statistical physics.