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anyone wanna talk about how applying the central limit theorem to a random walk in 2D reproduces the euclidean metric starting from a *grid* it’s fucked up and wrong and i’m still trying to figure out what it means

Emily

@the_aiju

random walks have a funny habit of introducing square roots

the square roots are whatever, what fucks me up is the random walk has a discrete symmetry and its scaling limit has a much bigger continuous symmetry

Neal Coleman

@necoleman

i get where the translations come from but where do the rotations come from?

you mean where do they come from in the limit? the limiting distribution (after appropriate scaling) is a gaussian with density something like exp(-x^2 -y^2) and that exponent is what i’m pointing at

Neal Coleman

@necoleman

I mean if I think of (eps)Z^2 as approximating R^2 (which is what you're doing when turning a random walk into brownian motion) and try to figure out how the symmetries of R^2 emerge in the limit, I can see where the translations come from, but I don't see where SO(2) comes from

6:02 PM · Sep 20, 2020Twitter for iPad

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and

oh yes if you’re saying what i think you’re saying then i agree, i don’t get it conceptually either. a priori you only get a D4 symmetry. there’s some stuff about brownian motion and harmonic functions that seems related and that i also don’t get

Replying to

and

i mean there's an approximate symmetry that becomes exact in the limit

Neal Coleman

@necoleman

given angle \theta is it easy to define a quasi isometry (\eps Z)^2\to (\eps Z)^2 that becomes a rotation by \theta in the limit?

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and

there's that stat mech connection you mentioned earlier! starting a particle from a point and letting it diffuse is basically the same* as evolving a dirac delta through the heat equation *modulo all sorts of exciting functional analysis