Functional renormalization group for p=2 like glassy matrices in the planar approximation: I. Vertex expansion at equilibrium

In this paper, we study the equilibrium states of a $N\times N$ stochastic complex random matrix $M$, whose entries evolve in time accordingly with a Langevin equation including both Gaussian white noises and a linear disorder, materialized by the Wigner random matrices. In large $N$-limit, the disorders behave as effective kinetics, and we examine a coarse-graining over the Wigner spectrum accordingly with two different schemes that we call respectively active and passive. We then investigate explicit solutions of the nonperturbative renormalization group using vertex and derivative expansion, a simple way to deal with the nonlocal nature of the effective field theory at large $N$. Our main statement is the existence of well-behaved fixed point solutions and at least some evidence about a discontinuous (first order) phase transition between a condensed and a dilute phase. We finally interpret the resulting phase space regarding the out-of-equilibrium process related to the dynamical phase transitions.