Fluxes, Laplacians, and Kasteleyn’s theorem

The following problem, which stems from the "flux phase" problem in condensed matter physics, is analyzed and extended here: One is given a planar graph (or lattice) with prescribed vertices, edges and a weight |t xy | on each edge (x, y). The flux phase problem (which we partially solve) is to find the real phase function on the edges, (x, y), so that the matrix T: = |t xy |expi (x, y) minimizes the sum of the negative eigenvalues of -T. One extension of this problem which is also partially solved is the analogous question for the Falicov-Kimball model. There one replaces the matrix -T by -T + V, where V is a diagonal matrix representing a potential. Another extension of this problem, which we solve completely for planar, bipartite graphs, is to maximize |det T |. Our analysis of this determinant problem is closely connected with Kasteleyn's 1961 theorem (for arbitrary planar graphs) and, indeed, yields an alternate, and we believe more transparent proof of it.