We present a new efficient method to find the Ising problem partition function for finite lattice graphs embeddable on an arbitrary orientable surface, with integral coupling constants bounded in the absolute value by a polynomial of the size of the lattice graph. The algorithm has been implemented for toroidal lattices using modular arithmetic and the generalized nested dissection method. The implementation has substantially better performance than any other as far as we know.

Kasteleyn stated that the generating function of the perfect matchings of a graph of genus $g$ may be written as a linear combination of $4^g$ Pfaffians. Here we prove this statement. As a consequence we present a combinatorial way to compute the permanent of a square matrix.

We compute the exact partition function of 2d Ising spin glasses with binary couplings. In these systems, the ground state is highly degenerate and is separated from the first excited state by a gap of size 4J. Nevertheless, we find that the low temperature specific heat density scales as exp(−2J/T ), corresponding to an "effective" gap of size 2J; in addition, an associated cross-over length scale grows as exp(J/T ). We justify these scalings via the degeneracy of the low lying excitations and by the way low energy domain walls proliferate in this model. In this work we reconsider the nature of these singularities using recently developed methods [6,7] for computing the exact partition function of square lattices with periodic boundary conditions, focusing on the low T scaling properties of the model with binary couplings. We show that although the energy "quantum" of excitation above the ground state is 4J, such excitations behave as composite particles; in fact the specific heat near the critical point scales as if the elementary excitations were of energy 2J. We justify this picture using properties of excitations and domain walls in this model. Finally, the joint temperature and size dependence shows the presence of a characteristic temperature-dependent length that grows as exp(J/T ), in agreement with hyperscaling.The model and our measurements -The Hamiltonian of our two-dimensional (2d) spin glass iswhere the sum runs over all nearest neighbor pairs of Ising spins (σ i = ±1) on a square lattice of volume V = L × L with periodic boundary conditions. The quenched random couplings J ij take the value ±J with probability 1/2. The partition function at inverse temperature β ≡ T −1 is Z J = {σi} e −βHJ ({σi}) and can be written asHere P J (X) is the polynomial whose coefficient of X p is the number of spin configurations of energy E = (−2L 2 + 2p)J. Saul and Kardar [4,5] showed that determining P J can be reduced to computing determinants which they did using exact arithmetic of arbitrarily large integers. More recently a more powerful approach has been developed [6,7], based on the use of modular arithmetic to compute pfaffians. With this algorithm, one first finds the coefficients modulo a prime number, thereby avoiding costly arbitrary precision arithmetic. Then the computation is repeated for enough different primes to allow the reconstruction of the actual (huge) integer coefficients using the Chinese remainder theorem.The algorithm proposed and implemented in [6,7] is powerful enough to solve samples with L ≈ 100; the total CPU time needed to compute Z J grows approximately as L 5.5 . In our study we have determined Z J for a large number of disorder samples at different lattice sizes: for instance we have 400000 samples at L = 6, 100000 at L = 10, 10000 at L = 30, 1000 at L = 40 and 300 at L = 50. The total computation time used is equivalent to about 40 years of a 1.2 GHz Pentium processor. For each sample we derive from Z J various thermodynamic quantities such as the free energy F J (β) = −β −1 ln Z...

Let H be a fixed graph. We show that any H-minor free graph G of high enough girth has circular chromatic number arbitrarily close to two. Equivalently, each such graph G admits a homomorphism to a large odd circuit. In particular, graphs of high girth and of bounded genus, or of bounded tree width, are``nearly bipartite'' in this sense. For example, any planar graph of girth at least 16 admits a homomorphism to a pentagon. We also obtain tight bounds on the girth of G in a few specific cases of small forbidden minors H. Academic Press

We present a polynomial time algorithm to find the maximum weight of an edge-cut in graphs embeddable on an arbitrary orientable surface, with integral weights bounded in the absolute value by a polynomial of the size of the graph.The algorithm has been implemented for toroidal grids using modular arithmetics and the generalized nested dissection method. The applications in statistical physics are discussed.