  We consider the complex branching random walk on a dyadic tree with Gaussian
weights on the boundary between the diffuse phase and the glassy phase. We
study the branching random walk in the space of continuous functions and
establish convergence in this space. The main difficulty here is that the
expected modulus of continuity of the limit is too weak in order to show
tightness in the space of continuous functions by means of standard tools from
the theory of stochastic processes.
