UBC Math Department Lecture: Tony Guttmann

There are a number of seminal two-dimensional lattice models that are integrable, but have only been partially solved, in the sense that only some properties are fully known (e.g. the two-dimensional Ising model, where the free-energy is known, but not the susceptibility). Alternatively, critical properties are known for some lattices but not others. For example, the critical point of the self-avoiding walk model is known rigorously for the honeycomb lattice, but not for other lattices. Similarly for the q-state Potts model and both bond and site percolation. The critical manifold of the former is known only for some lattices, likewise the percolation threshold is known only for some lattices.

A range of numerical procedures exist, based on exact enumeration, or other numerical work, such as calculating the eigenvalues of transfer matrices, which, when combined with various structural invariants seem to give exact results in those cases that are known to be exact, but can be used to give increasingly precise estimates in those cases which are not exactly known. Reasons for this partial success are not well understood. In this talk I will describe four such procedures, and demonstrate their performance, and speculate on their partial success.