Dimer Statistics on a Bethe Lattice

We discuss the exact solutions of various models of the statistics of dimer coverings of a Bethe lattice. We reproduce the well-known exact result for noninteracting hard-core dimers by both a very simple geometrical argument and a general algebraic formulation for lattice statistical problems. The algebraic formulation enables us to discuss loop corrections for finite dimensional lattices. For the Bethe lattice we also obtain the exact solution when either (a) the dimers interact via a short-range interaction or (b) the underlying lattice is anisotropic. We give the exact solution for a special limit of dimers on a Bethe lattice in a quenched random potential in which we consider the maximal covering of dimers on random clusters at site occupation probability p. Surprisingly the partition function for “maximal coverage” on the Bethe lattice is identical to that for the statistics of branched polymers when the activity for a monomer unit is set equal to −p. Finally we give an exact solution for the number of residual vacancies when hard-core dimers are randomly deposited on a one dimensional lattice.