A formula for boundary correlations of the critical Ising model

Abstract

Given a finite rhombus tiling of a polygonal region in the plane, the associated critical $Z$-invariant Ising model is invariant under star-triangle transformations. We give a simple matrix formula describing spin correlations between boundary vertices in terms of the shape of the region. When the region is a regular polygon, our formula becomes an explicit trigonometric sum.

21 pages, 5 figures; v2: Bibliography updates, section structure reorganized; v3: Final version to appear in Probab. Theory Related Fields