Renormalization theory of self-avoiding walks which cross a square

Abstract

The renormalization group theory is used to calculate the critical behaviour of self-avoiding walks which cross a square. This problem, which has been proposed recently, is especially well suited for a renormalization group analysis. The fixed-endpoint, diagonal-span trademark of the square-crossing walks leads naturally to a well defined renormalization scheme. Unlike other finite-lattice renormalization schemes for self-avoiding walks, this square-crossing renormalization is exact in the sense that the finite lattice (square) is uniquely defined, the spanning rule is unambiguous, and the end-to-end correlations are exactly preserved. The results for the critical point are in excellent agreement with series analysis estimates and support a conjecture on its exact value.