Tightness for branching random walk in time-inhomogeneous random environment
Xaver Kriechbaum
TL;DR
This work analyzes the maximum of a branching random walk in a time-inhomogeneous random environment and proves annealed tightness for the centered maximum M_n−m_n, where the centering m_n depends on the environmental sequence via barrier probabilities and a barrier-driven functional K_n. The authors develop barrier-based ballot arguments, a many-to-one lemma adapted to the environment, and a detailed barrier calculus to bound right- and left-tail probabilities; they further introduce a shifting-starting-point technique and a tree-cutting strategy to manage dependencies and obtain tightness. A substantial portion is dedicated to barrier computations, including barrier-splitting, Girsanov tilting, and dyadic decompositions to control error terms and constants, ensuring robustness across the environment. The results elucidate the asymptotic behavior of BRWRE maxima under Gaussian increments and provide a framework potentially extensible to more general environments, with implications for understanding universality and distributional limits in BRW in random environments.
Abstract
We consider a branching random walk in time-inhomogeneous random environment, in which all particles at generation $k$ branch into the same random number of particles $\mathcal{L}_{k+1}\ge 2$, where the $\mathcal{L}_k$, $k\in\mathbb{N}$, are i.i.d., and the increments are standard normal. Let $\mathbb{P}$ denote the law of $(\mathcal{L}_k)_{k\in\mathbb{N}}$, and let $M_n$ denote the position of the maximal particle in generation $n$. We prove that there are $m_n$, which are functions of only $(\mathcal{L}_k)_{k\in\{0,\dots, n\}}$, such that (with regard to $\mathbb{P}$) the sequence $(M_n-m_n)_{n\in\mathbb{N}}$ is tight with high probability.