What is renormalization group? Would love to hear how a Mathematician thinks about it as opposed to a Physicist.
Conversation
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.
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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
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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
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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"
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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?"
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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}
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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:
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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!
that's *sorta* the CLT in a nutshell, and even the convergence of a random walk to brownian motion if you squint harder
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Amazing!! I didn’t know about the RG flow approach to CLT but makes a lot of sense. It looks like to me that cumulants are really the key objects and they are somehow underestimated in math. Thank you!
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