# Percolation in a 2D Ising model

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?

• Yes your conjecture is right. The Ising model is connected to the bond-correlated percolation model. See this paper. May 4, 2012 at 9:33
• Thanks, though my suspicions were that it is related to site percolation. Specifically, I would expect that the spin configuration of the high temperature Ising model can be mapped to the configuration of sites for p \approx 1/2 bond percolation. I suppose that the bond-correlated percolation mapping also works at finite temperature. May 4, 2012 at 9:42
• @VijayMurthy it would be great if you post that as an answer (and just add a few words about the argument given in the paper). May 4, 2012 at 15:50

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.