Since yesterday I've been on a conference about functional #renormalization group in #physics , taking place in #Trento in Italy. Consequently I've been learning about many old and new papers on that topic. We start with something old. Monday's #paperOfTheDay is "Renormalization Group Equation for Critical Phenomena" from 1973. This is (one of) the very first papers to introduce what is now know as functional renormalization group methods, namely, the idea to solve the path integral by integrating out only one momentum shell at a time, while keeping track of an effective action that "flows" from the classical action to the full quantum effective action. This particular paper works with spins and also examines the limit N->infinity of the O(N) symmetric model.
What I found particularly interesting was the general argument why the right-hand side of such a flow equation can contain at most second derivatives of the effective action: This has to do with expectation values of products of many spins decaying quickly in the continuum limit. Also the tropical loop equation in #tropicalFieldTheory has second derivatives (as has the analogous equation for 0-dimensional QFT). There, this property was obvious from Feynman diagrams: Cutting a loop in a Feynman diagram amounts to cutting exactly one propagator, and every propagator has exactly two ends, therefore this necessarily produces a diagram with two more legs, hence a second derivative in the generating function.
https://journals.aps.org/pra/abstract/10.1103/PhysRevA.8.401