Simulation reduction of the Ising model to general matchings. (English) Zbl 1246.82018

Huber, Mark L.; Law, Jenny

Electron. J. Probab. 17, Paper No. 33, 15 p. (2012).

Summary: A distribution is tractable if it is possible to approximately sample from the distribution in polynomial time. Here, the ferromagnetic Ising model with unidrectional magnetic field is shown to be reducible to a standard distribution on matchings that is tractable. This provides an alternate method to the original Jerrum and Sinclair approach to show that the Ising distribution itself is tractable. Previous reductions of the Ising model to perfect matchings on different graphs exist, but these older distributions are not tractable. Also, the older reductions did not consider an external magnetic field, while the new reduction explictly includes such a field. The new reduction also helps to explain why the idea of canonical paths is so useful inapproximately sampling from both problems. In addition, the reduction allows any algorithm for matchings to immediately be applied to the Ising model. For instance, this immediately yields a fully polynomial time approximation scheme for the Ising model on a bounded degree graph with magnetization bounded away from 0, merely by invoking an existing algorithm for matchings.

MSC:

82B20	Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
60C05	Combinatorial probability
65C05	Monte Carlo methods
82B80	Numerical methods in equilibrium statistical mechanics (MSC2010)

Keywords:

Monte Carlo; simulation reduction; canonical paths; Ising model; fully polynomial time approximation scheme