# Condition for Phase transition

1. May 29, 2015

### JorisL

Hi

I'm trying to use Peierl's argument which in essence is clear to prove that there does exist a phase transition in the 2D Ising model without external field.
The issue I'm having is of a more mathematical nature, in class it was mentioned that there is a phase transition if for some $\delta > 0$, the probability $\mathbb{P}_V^+[\sigma_0 = -1] \leq \frac{1}{2}-\delta$ uniform as $V\uparrow\mathbb{Z}^2$.
Let me clarify the notation, we are looking at a finite subvolume V of the square lattice.
The superscript specifies the boundary conditions, all + in this case. And $\sigma_0$ is the state of the site at the origin.

I don't see why this is true, is it because in that case the average magnetisation is non-zero when going to the infinite lattice?
I've been able to follow through the Peierl's argument which is pretty nifty as is, but the sometimes very mathematical approach we used in the course makes it hard to couple back to the examples.

Thanks,

Joris

2. May 29, 2015

### JorisL

I have confirmed that this exactly justifies the claim.
I hate moments like these when something seems too good/simple to be true.

Joris