Maxima of branching random walks with piecewise constant variance

Abstract

This article extends the results of Fang & Zeitouni (2012a) on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in [0,1] under a slight restriction. We find the asymptotics of the maximum up to an OP(1) error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein (2015b) when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang & Zeitouni (2012a) instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.

28 pages, 4 figures

References in corpus (10)

• Maximum of the characteristic polynomial of random unitary matrices
• Minima in branching random walks
• Tightness for a family of recursion equations
• Late points for random walks in two dimensions
• Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment
• Variable speed branching Brownian motion 1. Extremal processes in the weak correlation regime
• A note on the maximum of the Riemann zeta function, and log-correlated random variables
• Multiplicative chaos measures for a random model of the Riemann zeta function
• Second order term of cover time for planar simple random walk
• Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Cited by in corpus (3)

• Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Sub-leading order and exponential tails
• Tightness for branching random walk in time-inhomogeneous random environment
• From 1 to infinity: The log-correction for the maximum of variable-speed branching Brownian motion