Upper moderate deviation probabilities for the maximum of a branching random walk

TL;DR

The article advances the understanding of the maximal displacement in branching random walks by characterizing upper moderate deviations in the regime where the excess height $x_n$ grows to infinity no faster than $\sqrt{n}$. It develops a refined barrier-truncation strategy anchored in spinal decomposition and random-walk renewal theory to obtain precise asymptotics for $\mathbb{P}(M_n>m_n+x_n)$, including lattice and non-lattice settings, and identifies universal constants via limits of integrals involving the tail of $M_\ell$. By isolating non-contributing trajectories and proving asymptotic equivalences with controlled moments, the work also connects to the extremal process and yields a law of the maximum in a two-speed BRW in the mean regime, revealing a Gumbel limit with a random shift. These results refine the known upper-deviation bounds, unify lattice and non-lattice behavior, and provide tools for analyzing decorrelated extremal structures in spatial branching processes with potential applications to related stochastic growth models.

Abstract

Consider $M_n$ the maximal position at generation $n$ of a supercritical branching random walk. Aïdékon (2013) obtained and described the convergence in law, as time $n$ goes to infinity, of $M_n-m_n$, where $m_n$ is an explicit function. Equivalently, he identified the limit of $\mathbb{P}(M_n > m_n + x)$, for any $x \in \mathbb{R}$. More recently, Luo (2025) gave an asymptotic equivalent for the upper large deviation probability, that is $\mathbb{P}(M_n > m_n + xn)$, for $x > 0$. In this work, we study an intermediate regime, called upper moderate deviation. We obtain, under close-to-optimal integrability conditions, an asymptotic equivalent for $\mathbb{P}(M_n > m_n + x_n)$, where $x_n$ is such that $x_n \to \infty$ and $x_n = O(\sqrt{n})$. Our proof is based on a strategy due to Bramson, Ding, and Zeitouni (2016). As a byproduct, we obtain information about the typical behavior of particles contributing to such deviations. Finally, we apply our main result to show the convergence in law of the centered maximum of a two-speed branching random walk in the mean regime and describe its limit.

Theorems & Definitions (65)

• Theorem 1.1
• Remark 1.2
• Remark 1.3
• Corollary 1.4
• Corollary 1.5
• Conjecture 1.6
• Lemma 2.1: Many-to-one formula, Theorem 1.1 of Shi2015
• Theorem 2.2: Lyons' spinal decomposition theorem Lyons1997
• Lemma 2.3: Doney Doney2012
• Lemma 2.4: Pain Pain2018