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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"

1:56 AM · Jan 27, 2021Twitter Web App

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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

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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?"