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We consider the complex branching random walk on a dyadic tree with Gaussianweights on the boundary between the diffuse phase and the glassy phase. Westudy the branching random walk in the space of continuous functions andestablish convergence in this space. The main difficulty here is that theexpected modulus of continuity of the limit is too weak in order to showtightness in the space of continuous functions by means of standard tools fromthe theory of stochastic processes.

Continuity estimates for the complex cascade model on the phase boundary