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A universality class is an equivalence class of physical models – field theories, quantum field theories, or models of classical or quantum statistical physics – where the equivalence is defined by two or several models' having the same mathematical description of the behavior at very long time scales and distance scales. So if two models' behavior at very ...

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You can think of the whole thing as a "fluid of systems" and each one of them can be in any of the states $i$ available. $\pi_{ij}$ tells you what is the speed at which a system in state $i$ will go to state $j$ $p_i$ tells you how likely it is for a system (in this fluid or ensemble of systems) to be in state $i$ and is proportional to the number of ...

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As you say, the correlation length, $\xi$, is a measure of domain size. Two spins that are within a correlation radius will have similar statistics. A system where there is such a scale can not be scale invariant. Indeed, scale invariance means that you can zoom in or out of the system and finds that it still looks the same. When there is a correlation ...

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That is exactly right. I would say "exactly one spin up" rather than "any spin up", but that is just phrasing. Another way of looking at this is that there is an increase in entropy (of $\log n$ times Boltzman's constant) associated with flipping one spin, which will sometimes overcome the energy cost.

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For the 2D planar Ising Edward-Anderson model, it can be solved in polynomial time. For the 3D case it is NP-hard, moreover NP-Complete.[1] [1] Barry A. Cipra, The Ising Model Is NP-Complete

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