The extremal process of two-speed branching random walk

TL;DR

This work analyzes the extremal behavior of a two-speed branching random walk in a time-inhomogeneous environment, establishing a discrete analog of the two-speed BRW results known for branching Brownian motion. The authors prove that in the fast regime the centered maximum converges to a Gumbel distribution with a random shift and the extremal process converges to a randomly shifted decorated Poisson point process, with the shift driven by the additive martingale limit of the first environment. In the slow regime they obtain a similar, but conditioning-driven, limit where the derivative martingale from the first environment governs the random intensity of the limiting point process, and the decorations derive from the second environment; a mean-regime conjecture is also proposed. The proofs combine spinal decomposition, additive and derivative martingales, and existing results on extremal processes, providing a unified framework for understanding inhomogeneous BRWs and their extremal statistics.

Abstract

We consider a two-speed branching random walk, which consists of two macroscopic stages with different reproduction laws. We prove that the centered maximum converges in law to a Gumbel variable with a random shift and the extremal process converges in law to a randomly shifted decorated Poisson point process, which can be viewed as a discrete analog for the corresponding results for the two-speed branching Brownian motion, previously established by Bovier and Hartung [12].

Theorems & Definitions (37)

• Theorem 1.1: Fast regime
• Remark 1.1
• Theorem 1.2: Slow regime
• Conjecture 1.3: Mean regime
• Theorem 2.1: Lyonslyons
• Corollary 2.2: Theorem 1.1 in shi
• Theorem 2.3
• Remark 2.1
• Lemma 2.4
• proof