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

Biggins [Uniform convergence of martingales in the branching random walk. {\em 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. There may be a part of the boundary, on which the martingales do not exist, on other parts it exists, but diverges or vanishes in the limit. In the remaining part, there is convergence to a non-degenerate limit. The arguments that give this convergence also apply in $\Lambda$ and require weaker moment assumptions than the ones used by Biggins.