Minima in branching random walks

Abstract

Given a branching random walk, let $M_n$ be the minimum position of any member of the $n$th generation. We calculate $\mathbf{E}{M}_n$ to within O(1) and prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}{M}_n|>x\}$, under quite general conditions on the branching random walk. In particular, together with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results fully characterize the possible behavior of $\mathbf{E}{M}_n$ when the branching random walk has bounded branching and step size.

Published in at http://dx.doi.org/10.1214/08-AOP428 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Tightness for a family of recursion equations

Cited by in corpus (33)

• Convergence in law of the minimum of a branching random walk
• Maximum of the characteristic polynomial of random unitary matrices
• Extreme local extrema of two-dimensional discrete Gaussian free field
• A simple path to asymptotics for the frontier of a branching Brownian motion
• The Seneta--Heyde scaling for the branching random walk
• Slowdown for time inhomogeneous branching Brownian motion
• Exact asymptotics of the freezing transition of a logarithmically correlated random energy model
• Maximal displacement of a branching random walk in time-inhomogeneous environment
• High values of disorder-generated multifractals and logarithmically correlated processes
• Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment
• Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees
• Maximal displacement in the $d$-dimensional branching Brownian motion
• Poisson-Dirichlet branching random walks
• The precise tail behavior of the total progeny of a killed branching random walk
• On explosions in heavy-tailed branching random walks
• Precise tail asymptotics of fixed points of the smoothing transform with general weights
• Genealogy of the extremal process of the branching random walk
• On the overlap distribution of branching random walks
• Maxima of branching random walks with piecewise constant variance
• Longest path distance in random circuits
• Extremes of Multi-type Branching Random Walks: Heaviest Tail Wins
• Branching-stable point measures and processes
• Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate
• Right-Most Position of a Last Progeny Modified Branching Random Walk
• Right-Most Position of a Last Progeny Modified Time Inhomogeneous Branching Random Walk
• Effective resistance of random trees
• First passage percolation in hostile environment is not monotone
• The largest fragment of a homogeneous fragmentation process
• Large deviations and the emergence of a logarithmic delay in a nonlocal linearised Fisher-KPP equation
• Last passage percolation and limit theorems in Barak-Erdős directed random graphs and related models
• Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk
• Shape theorem for a one-dimensional growing particle system with a bounded number of occupants per site
• Extremal Process of Last Progeny Modified Branching Random Walks