# What is rigorously known about critical points?

+ 3 like - 0 dislike
169 views

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

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.
Anyway, 4D Ising is unphysical since the physically relevant dimension is $d=3$.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.