Convergence of complex martingales in the branching random walk: the boundary

Konrad Kolesko,Matthias Meiners

doi:10.1214/17-ecp50

Abstract

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda $ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda $ of $\Lambda $. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda $ and require weaker moment assumptions than the ones used by Biggins.

Highlights

Biggins [8] proved local uniform convergence of additive martingales in a supercritical branching random walk on Rd at complex parameters within a certain open set Λ ⊆ Cd
The arguments that give convergence to a non-degenerate limit apply in Λ and require weaker moment assumptions than the ones used by Biggins
Besides its value in the study of large deviation results for the branching random walk and its intrinsic interest, there is further motivation to study the convergence of additive martingales at complex parameters, on the boundary ∂Λ

Summary

Introduction

Biggins [8] proved local uniform convergence of additive martingales in a supercritical branching random walk on Rd at complex parameters within a certain open set Λ ⊆ Cd. In each known example and in the whole setup of [20], the main result of which is the description of the set of all solutions to non-critical smoothing equations, this solution can always be chosen as the limit of an additive martingale in a suitable branching random walk at a complex parameter from Λ. If one aims at extending the results from [20] from non-critical to critical smoothing equations, limits of additive martingales in suitable branching random walks at complex parameters from the boundary of Λ figure. Theorem 2.1, might be viewed as a discrete, non-Gaussian counterpart of the convergence of the suitably normalized approximations to complex Gaussian multiplicative chaos measures in the diffuse phase and the boundary between the diffuse and the glassy phase

Main results

Discussion and examples

Proofs of the main results

Results for higher dimensions and the connection to smoothing transforms

A Auxiliary results