Imitation of 2d quantum field theory by means of REM like models

An imitation of 2d field theory is formulated by means of a model on the hierarchic tree (with branching number close to one) with the same potential and the free correlators identical to those of 2d ones. Such a model possesses some features of original models for certain scale invariant theories. For the case of 2d conformal models it is possible to derive exact results. The renormalization group equation for the free energy is a reaction-diffusion equation, which is noise-free KPZ equation with an additional linear term. For the case of Liouville model and strings these models on trees may be naturally expressed via the Random Energy Model. This correspondence is used to identify the phase structure of strings for analytical continuation of DDK expressions. A phase transition is found for spherical strings a bit below three dimensions.