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What is renormalization group? Would love to hear how a Mathematician thinks about it as opposed to a Physicist.

conveniently i was just reading about this! you should really read david tong's notes on statistical field theory but i'll attempt a summary in relatively mathematical language of what i've gotten out of it so far: damtp.cam.ac.uk/user/tong/sft.

so, there's some process (statistical mechanics / statistical field theory) by which we go from a microscopic description of individual particles to a macroscopic description of how a bunch of particles behave together. the ising model is currently my favorite toy model of this

that process pretty much has to have the property that it's relatively insensitive to scaling up: that is, if i know how a brick of molecules behaves, i also know how a brick twice as large in each dimension behaves, just some parameters (e.g. mass, volume) change

under nice circumstances you can describe a macroscopic system in terms of some parameters in such a way that scaling the system corresponds to a simple change in the parameters. this is "renormalization group flow"

and (this is the part of the argument i'm shakiest on the details) what we see macroscopicaly should therefore be approximately a fixed point of RG flow. the simplest toy model i've found for understanding this is the central limit theorem

the "RG proof of CLT" goes like this: you take a random walk (adding i.i.d. copies of a random variable to itself over and over again) and ask "how does this behave if i let it run a long time and zoom out a lot?"

the "RG flow" here is a "blocking transformation" where instead of looking at (X_1 + ... + X_n) / sqrt{n} you instead consider the effect of "fuzzing" out each step X_i into a pair of steps (X_i + X_{i+1}) / sqrt{2}

this naturally leads you to investigate the map X_i -> (X_i + X_{i+1}) / sqrt{2} on (mean-zero) random variables, and if you look at what this map does to cumulants (the coefficients of the log of the characteristic function / moment generating function) you see the following:

2:01 AM · Jan 27, 2021Twitter Web App

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the second cumulant, which is the variance, doesn't change. the other higher cumulants decay exponentially - in RG terms these are "irrelevant directions" and in dynamical systems terms this is the "stable manifold"

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so we see immediately that the only fixed point of "RG flow" is the one with the second cumulant nonzero and all other cumulants zero, which is exactly a gaussian, and moreover it's an attractive fixed point!