Polymer collapse transition: a view from the complex fugacity plane

Distributions of zeros of the grand canonical and canonical partition functions in the complex fugacity and in the complex interaction strength plane are examined numerically for a model of rooted self-interacting self-avoiding walks on a hierarchical graph. It is shown that the pattern of zeros of the grand canonical partition function in the complex fugacity plane has a circular-like form, with the exception of zeros lying in the vicinity of the critical point. Exact values of polymer size critical exponents, in the swollen and dense phase, as well as at the point of collapse transition, are obtained by studying asymptotic behavior of zeros lying closest to the positive real axis in the fugacity plane. Distribution of zeros of the canonical partition function in the complex interaction strength plane is found to be more complicated, and an accurate estimate of crossover exponent is obtained from study of asymptotic behavior of zeros lying closest to the positive real axis in this plane. It is also shown that the next-to-leading term in the asymptotic law for canonical partition function in the dense phase has a stretched-exponential rather than the power-law form.