Evaluation of the propagator of Susskind fermions by summation of non-reversal random walks

Abstract

We consider the Susskind action for quarks on the lattice which possesses a chiral invariance as the quark mass m goes to zero. The propagator G( y, x) in a gluon field U may be expanded in a series whose terms correspond to all random walks from x to y with hopping parameter κ = m −1. It is observed that a step followed by its reversal (a spike) is independent of U because of factors UU −1. It follows that the contribution from the insertion of all possible trees (made of spikes) at all sites of each walk may be evaluated by simple combinatorics. The propagator in a generic gluon field U is thereby expressed as a sum over all non-reversal walks with renormalized hopping parameter κ′ = 2κ/[1 + √(1 + 28κ 2)] < √ 1 7 (⪡ κ for small m) . The observation that non-reversal random walks define a Markov chain provides a new expression for the propagator as the inverse of a new matrix. The spectrum of the new matrix is found for generic U. It is shown that the new expression for the propagator may be evaluated by an iteration procedure with ratio of successive errors bounded by 1 − 1 8 √ 1 7 m + O(m 2) , whereas the corresponding bound for the original expression is 1− 1 128 m 2 + O(m 4) . The advantage of the new expression for small m is obvious.