New method for analyzing confluent singularities and its application to two-dimensional percolation

Abstract

Several persistent discrepancies between different results for critical exponents of statistical-mechanical models have recently been shown to originate from inappropriate treatment of confluent singularities. In this paper a new method for evaluating ${\ensuremath{\Delta}}_{1}$, the critical exponent of the confluent correction term, from relatively short series expansions is introduced and applied to $d=2$ isotropic percolation to obtain estimates of both ${\ensuremath{\Delta}}_{1}$ and $\ensuremath{\gamma}$. The discrepancies noted in the literature between values of $\ensuremath{\gamma}$ are clearly shown to be removed by the application of our method.