Biased branching random walks on Bienaymé--Galton--Watson trees

TL;DR

This work analyzes a $\lambda$-biased branching random walk on Bienaymé--Galton--Watson trees, focusing on the linear growth of the particle cloud. By linking the maximal (and, in a transient regime, minimal) displacement to the large deviation rate function $I_{\lambda}$ of a single $\lambda$-biased walk, the authors identify the speeds $v_{\lambda,m}$ and $\tilde{v}_{\lambda,m}$ through the relations $I_{\lambda}(a) \le \log m$. They establish a recurrence/transience dichotomy and prove zero-one laws to upgrade positive-probability events to almost-sure behavior for BGW environments, using annealed analyses and decoupling arguments via branching processes in random environments. The results extend shape-theorem type conclusions to branching Markov chains on random trees and connect the geometry of the BGW environment with large-deviation principles, offering a rigorous description of front propagation in this random setting.

Abstract

We study $λ$-biased branching random walks on Bienaymé--Galton--Watson trees in discrete time. We consider the maximal displacement at time $n$, $\max_{\vert u \vert =n} \vert X(u) \vert$, and show that it almost surely grows at a deterministic, linear speed. We characterize this speed with the help of the large deviation rate function of the $λ$-biased random walk of a single particle. A similar result is given for the minimal displacement at time $n$, $\min_{\vert u \vert =n} \vert X(u) \vert$.

Figures (4)

• Figure 1: The different recurrence and transience regimes for the $\lambda$--biased random walk and $\lambda$--biased branching random walk.
• Figure 2: The spectral radius $\rho_\omega(\lambda)$ is determined by atypical regions of $\omega$: the set $A(x,2L)$ consists of $L$ levels of a $d^{(1)}$--ary tree and $L$ levels of a $d^{(2)}$--ary tree. The choice of $(d^{(1)},d^{(2)})$ depends on $\lambda$: if $\lambda < d_{\min}$ we choose $d^{(1)}=d^{(2)}=d_{\min}$ and if $d_{\min} \leq \lambda < m_{\mathtt{BGW}}$ we choose $d^{(1)}=d_0$ and $d^{(2)}=d_{\min}$ where $d_0>m_{\mathtt{BGW}}$. Then the set $A(x,2L)$ acts as a trap for the random walk. Similarly, for the branching random walk, $A(x,2L)$ acts as a seed which facilitates local survival.
• Figure 3: An illustration of the event $\mathtt{split}(x)$: the tagged particle $S^*$ (in solid red) visits $x$ but not $F_\omega(x)\backslash \{x\}$ (in light grey). While $S^*$ visits $x$, the BRW produces a new particle $v$ (in dashed green) that moves to a child of $x$ and then never revisits $x$.
• Figure 4: An illustration of the trajectories counted in $\mathcal{Z}_i^{(n)}$. Observe that the trajectories are allowed to backtrack but not below level $D_{(i-1)n}$ so that the trajectories in different cones are independent. Under $\mathbb{P}$ the grey cones are independent copies of $\omega$. Each cone may have multiple particles starting in it, we compare this to a process where all particles use the same cone.

Theorems & Definitions (35)

• Theorem 1.1: Velocity of the maximal displacement
• Theorem 1.2
• Theorem 1.3: Velocity of the minimal displacement
• Proposition 1.4
• Proposition 1.5
• Remark 1.6
• Theorem 2.1: Existence of speed lyons_biased_1996
• Theorem 2.2: Large deviations, speed-up probabilities DGPZ-ld
• Theorem 2.3: Large deviations, slow-down probabilities DGPZ-ld
• Proposition 2.4