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

TL;DR

This paper explores the connection between Wilson RG, Wegner-Morris Fokker-Planck RG, and Carosso field-coarsening via Langevin stochastic processes, offering new perspectives on RG flows and their stochastic interpretations.

Contribution

It introduces a stochastic formulation of RG flows using Langevin equations, linking different RG schemes and analyzing large deviation properties of extensive observables.

Findings

Reformulation of RG flows as Langevin stochastic processes.

Connection between Carosso RG and stochastic heat equation.

Analysis of large deviation properties of the empirical magnetization.

Abstract

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 $E_t[\phi(.)]$ translates for the probability distribution $P_t[\phi(.)] \sim e^{- E_t[\phi(.)]} $ 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 $\phi_t(\vec x)$ 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 \equiv \frac{1}{L^d} \int_{L^d} d^d \vec x \ \phi(\vec 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 $L^d$.