If you want to find the ground state of the 2d random bond Ising model (no field), a computationally efficient algorithm exists to do it for you (based on minimum weight perfect matching). What about the thermal state for some finite temperature, T? Is there an efficient (i.e. time scales polynomially with system size) way to sample states of different spin configurations from this distribution?
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$\begingroup$ Does this help? arxiv.org/abs/cond-mat/0703043 $\endgroup$– wscCommented Jan 26, 2013 at 18:35
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The divide between tractable and intractable was shown by Sorin Istrail to be between planar and non-planar graphs. Since 2D surfaces can have non-planar graphs imbedded in them I think (correct me if I'm wrong) planar RBIMs are what you want to look at.
That said, Istrail showed that for {-J,0,J} couplings the ground state as well as the partition function can be efficiently calculated. I'm not sure of the exact details, but the calculation of the partition function can be mapped to the calculation of the determinant which is efficient. The specifics are available as references in the paper below.
Istrail's Paper: http://www.cs.brown.edu/people/sorin/pdfs/Ising-paper.pdf
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2$\begingroup$ The basic result, i.e.: that obtaining the random-bond-Ising GS for non-planar lattices is NP-complete, was proved by Barahona in 1982. www.yaroslavvb.com/papers/barahona-on.pdf $\endgroup$ Commented Jan 31, 2012 at 18:11