Exact transfer-matrix enumeration and critical behaviour of self-avoiding walks across finite strips

We use a transfer matrix with suitably defined vertex weights to algebraically enumerate n-step self-avoiding walks confined to cross an L*M rectangle on the square lattice. We construct the exact generating functions for self-avoiding walks from the south-west to south-east corners for L=1, 2, 3, 4, 5 and infinite height M corresponding to a half-strip. We also consider the number of n-sided polygons rooted to the south-west corner of the half-strip and give a formulation to treat self-avoiding walks across the full strip. In each case, the exact generating functions are ratios of polynomials in the step fugacity. We investigate the singularity structure of the generating functions along with the finite-size scaling in M of the singularity in the analogue of the heat capacity. We find the critical exponents gamma =1 and gamma =2 for the half- and open-strip, along with nu = 1/2 . These results are indicative of the one-dimensional or Gaussian nature of self-avoiding walks in infinitely long, but finitely wide strips.