  We report on our recent rigorous construction of complete renormalization
group trajectories between two fixed points for the three-dimensional phi-four
model with modified propagator considered by Brydges, Mitter and Scoppola
(BMS). These are discrete critical trajectories which connect the ultraviolet
Gaussian fixed point to the nontrivial BMS infrared fixed point which is an
analogue of the Wilson--Fisher fixed point. The renormalization group map is
defined rigorously and nonperturbatively, without using the hierarchical
approximation. The trajectories are constructed by a fixed point argument in a
suitable Banach space of sequences, where one perturbs a nonlinear
one-dimensional iteration.
