Journey from the Wilson exact RG towards the Wegner–Morris Fokker–Planck RG and the Carosso field-coarsening via Langevin stochastic processes

Within the Wilson RG of ‘incomplete integration’ as a function of the effective RG-time t, the non-linear differential RG-flow for the energy E t [ ϕ ( . ) ] translates for the probability distribution P t [ ϕ ( . ) ] ∼ e − E t [ ϕ ( . ) ] into the linear Fokker–Planck RG-flow associated to independent non-identical Ornstein–Uhlenbeck processes for the Fourier modes. The corresponding Langevin stochastic differential equations for the real-space field ϕ t ( x → ) have been recently interpreted by Carosso as genuine infinitesimal coarsening-transformations that are the analog of spin-blocking, and whose irreversible character is essential to overcome the paradox of the naive description of the Wegner–Morris Continuity-equation for the RG-flow as a meaningless infinitesimal change of variables in the partition function integral. This interpretation suggests to consider new RG-schemes, in particular the Carosso RG where the Langevin SDE corresponds to the stochastic heat equation also known as the Edwards–Wilkinson dynamics. After a pedestrian self-contained introduction to this stochastic formulation of RG-flows, we focus on the case where the field theory is defined on the large volume L^d with periodic boundary conditions, in order to distinguish between extensive and intensives observables while keeping the translation-invariance. Since the empirical magnetization m e ≡ 1 L d ∫ L d d d x →   ϕ ( x → ) is an intensive variable corresponding to the zero-momentum Fourier coefficient of the field, its probability distribution p L ( m e ) can be obtained from the gradual integration over all the other Fourier coefficients associated to non-vanishing-momenta via an appropriate adaptation of the Carosso stochastic RG, in order to obtain the large deviation properties with respect to the volume L^d .