Functional renormalization group for multilinear disordered Langevin dynamics I: Formalism and first numerical investigations at equilibrium

This paper aims at using the functional renormalization group formalism to study the equilibrium states of a stochastic process described by a quench--disordered multilinear Langevin equation. Such an equation characterizes the evolution of a time-dependent $N$-vector $q(t)=\{q_1(t),\cdots q_N(t)\}$ and is traditionally encountered in the dynamical description of glassy systems at and out of equilibrium through the so-called Glauber model. From the connection between Langevin dynamics and quantum mechanics in imaginary time, we are able to coarse-grain the path integral of the problem in the Fourier modes, and to construct a renormalization group flow for effective Euclidean action. In the large $N$-limit we are able to solve the flow equations for both matrix and tensor disorder. The numerical solutions of the resulting exact flow equations are then investigated using standard local potential approximation, taking into account the quench disorder. In the case where the interaction is taken to be matricial, for finite $N$ the flow equations are also solved. However, the case of finite $N$ and taking into account the non-equilibrum process will be considered in a companion investigation.