# What is rigorously known about critical points?

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What is rigorously known about the existence and properties of critical points (in the thermodynamic/statistical mechanics sense) in classical and quantum mechanical models in 3 space dimensions? I'd be particularly interested in pointers to survey articles that allow me to form a complete picture.

edited Sep 5, 2017

What do you mean by critical points? That which is called equilibria ($dH=0$) by Arnold? For that, I found the textbook Mathematical aspects of classical and celestial mechanics  by Arnold and others to be at the right briefness/detail ratio. Depending on what you are actually looking for, Bifurcation Theory and Catastrophe Theory from this series might also be interesting.

You probably already know this. I heard the proof for triviality of $\phi^4$ in 4D is almost rigorous, this implies the universality class for Ising model in 4D being Gaussian is at least almost rigorously proven (although I  don't know how difficult it is to have a rigorous justification for saying the two are indeed of the same universality). Saying mean field treatment for 4D Ising model critical exponent is exact is an equivalent statement, but I'm ignorant on how much rigor has been achieved from this perspective.

@JiaYiyang: Without a reference I don't buy the triviality proof. The arguments I know of depend on the assumption that you can get the field theory in a lattice limit, which is a questional assumption in the absence of asymptotic freedom. Thus what is to be proved is almost assumed from the start! -

See the discussion in https://www.physicsoverflow.org/32756 . The discussion in the paper by Gallavotti and Rivasseau from 1984 mentioned there is still unsurpassed, as far as I know. Se also  https://www.physicsoverflow.org/21328/

Anyway, 4D Ising is unphysical since the physically relevant dimension is $d=3$.

For the classical three-dimensional Ising model, it is rigorously known that the magnetization is continuous at $T_c$ and that the magnetic susceptibility diverges as $T\downarrow T_c$ (the actual result is stronger than that). A proof can be found in the paper https://arxiv.org/pdf/1311.1937.pdf. That still seems to be the state-of-the art. Note that the arguments are very specific to the Ising model (they rely crucially on the random-current representation).
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