Analysis of series expansions for non-algebraic singularities

Abstract

Existing methods of series analysis are largely designed to analyse the structure of algebraic singularities. Functions with such singularities have their $n^{th}$ coefficient behaving asymptotically as $A\cdot {\mu }^n\cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficients behave asymptotically as $B\cdot {\mu }^n\cdot {\mu }_1^{n^{\sigma }}\cdot n^g,$ where typically $\sigma =\frac{1}{2}$ or $\frac{1}{3}.$ Identifying this behaviour, and then extracting estimates for the critical parameters $B,\mu ,{\mu }_1,\sigma ,\mathrm{and}g$ presents a significant numerical challenge. We describe methods developed to meet this challenge.

arXiv comment35 pages, 11 figures (3 small typos corrected)

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• Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices
• A sharp asymptotics of the partition function for the collapsed interacting partially directed self-avoiding walk
• Compressed self-avoiding walks in two and three dimensions
• Progressive and Rushed Dyck Paths