For a 2D Ising model with zero applied field, it seems logical to me that the phases above and below T_c will have different percolation behaviour. I would expect that percolation occurs (and hence there is a spanning cluster) below T_c and that it does not occur (no spanning clusters) above T_c. I'm looking for conformation that this is true, but haven't found anything so far. Is my conjecture correct?
Statistical Mechanics – Analyzing Percolation in a 2D Ising Model
ising-modelpercolationstatistical mechanics
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I think you have a problem with double counting. The Ising Hamiltonian is $$H = - J \sum_{\langle i,j \rangle} S_i S_j$$ where this strange sum notation means to sum over all bonds between neighbouring spins. It is the bond that matters, so you should not count it twice (for $ij$ and $ji$). Actually you can get the energy difference of a spin flip as $$\Delta E = 2 J (N_{\uparrow\uparrow} - N_{\uparrow\downarrow})$$ where $N_{\uparrow\uparrow}$ ($N_{\uparrow\downarrow}$) is the number of neighbouring spins pointing in the same (opposite) direction before the flip.
The first thing to realize is that there are no "true" phase transitions (in the sense of non-analytic behaviour of thermodynamic potentials) in finite systems. This is the main difficulty one faces when analysing phase transitions using (most) computer simulation schemes.
In particular, such simulations are only reliable as long as the observed correlation length is significantly smaller than the system's linear size. However, when there is a second-order phase transition, the correlation length diverges at the critical point, which implies that close to the "true" critical temperature, the behavior observed in a finite system will be smoothed out (and, it turns out, the natural finite-volume analogue of the critical point is shifted, see below).
Now, of course, a large enough finite system will still display a behaviour that "resembles" a phase transition, but with its singularities smoothed out. To extrapolate results to infinite systems then requires (i) the determination of finite-volume analogues of the limiting quantities (in particular the critical temperature), (ii) examining how these finite-volume quantities change when the system's size is increased. In order to help with this extrapolation procedure, physicists have devised various finite-size scaling theories.
I assume that you are working on a torus (i.e., with periodic boundary conditions) of linear size $L$. This is the simplest case, as far as finite-size effects are concerned, since one then avoids the additional difficulties related to the presence of the system's boundary.
The first detailed finite-size scaling theory was developed by Ferdinand and Fisher in 1969 in a classical paper published in Phys. Rev. 185, 832. They used the exact results available for the two-dimensional Ising model to analyze finite-size effects on the free energy and specific heat.
The specific heat of a finite-volume Ising model does not diverge. However, it still displays a sharp increase in a narrow region around the "true" critical point $T_c$. Fisher and Ferdinand proposed to define the finite-volume analogue $T_c(L)$ of the critical temperature as the value of the temperature at which the specific heat is maximal. They then argued that $$ T_c - T_c(L) \sim L^{-1/\nu}\,, $$ where $\nu$ is the critical exponent associated to the specific heat, which is given by $\nu=1$ for the two-dimensional model.
Best Answer
You conjecture is correct. One can relate the 2d Ising model with the bond correlated percolation model. The details are in the paper Percolation, clusters, and phase transitions in spin models.
The basic idea is to consider interacting (nearest neighbor) spins as forming a bond with a certain probability. One can then show that the partition function of the Ising model is related to the generating function of the bond-correlated percolation model.
The above paper demonstrates that the bond-correlated percolation model has the same critical temperature and critical exponents as the 2d Ising model. However, the values of $T_c$ and the critical exponents seem to be dependent on exactly how one defines a bond. See section III.A.1 in Universality classes in nonequilibrium lattice systems (or arxiv version).
Nonetheless your intuitive picture that there would be spanning clusters below $T_c$ and no such clusters above $T_c$ remains valid.
EDIT 21 May 2012
I found a pedagogical paper that discusses this issue.