High precision canonical Monte Carlo determination of the growth constant of square lattice trees.

Abstract

The number of lattice bond trees in the square lattice (counted modulo translations), t(n), is a basic quantity in lattice statistical mechanical models of branched polymers. This number is believed to have asymptotic behavior given by t(n) approximately Alambda(n)n(-theta), where A is an amplitude, lambda is the growth constant, and theta the entropic exponent. In this paper, we show that lambda and theta can be determined to high accuracy by using a canonical Monte Carlo algorithm; we find that lambda=5.1439+/-0.0025, theta=1.014+/-0.022, where the error bars are a combined 95% statistical confidence interval and an estimated systematic error due to uncertainties in modeling corrections to scaling. If one assumes the "exact value" theta=1 and then determines lambda, then the above estimate improves to lambda=5.143 39+/-0.000 72. In addition, we also determine the longest path exponent rho and the metric exponent nu from our data: rho=0.74000+/-0.00062, nu=0.6437+/-0.0035, with error bars similarly a combined 95% statistical confidence interval and an estimate of the systematic error.