Model fermion Monte Carlo method with antithetical pairs

We consider certain model systems within which we seek solutions of the Schrödinger equation that are antisymmetric on inversion in the origin. We find geometrical constructions that define sequences of correlated random walks for which the overlap with antisymmetrical test functions are asymptotically constant. The emphasis here is on problems in two dimensions, but the methods generalize to many dimensions and to certain other model problems.