N ov 2 00 1 Imitation of 2 d quantum field theory by means of REM like models

Imitation of 2d quantum field theory by means of REM like models. Abstract 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. 1 Hierarchic tree with branching number close to 1. One of the most fruitful ideas in physics is the idea of universality. In fact, that the only hope due to which rather artificial models of the present theoretical physics can successfully capture the relevant aspects of the nature. We believe, that at the critical point the statistical mechanical system omits the secondary details. Usually this concerns the Hamiltonian in the d dimensional Euclidean space. It is proposed while keeping the Hamiltonian fixed to simplify the space geometry as much as possible retaining two point correlators and three point (for isosceles triangles) correlators. If the action of initial theory consists of the Laplacian and a potential, our model feels the space dimension through 1