Asymptotic behaviour of the perturbation expansion in n=0 field theories

Abstract

The author develops a method for studying the high-order behaviour of the perturbation expansion in theories in which the number of field components, n, is taken to zero. This procedure is illustrated on the field theory formulation of the percolation problem, which can be considered as the n=0 limit of the (n+1)-state Potts model. The saddle points controlling the asymptotic behaviour are labelled by an integer r=1, 2, . . ., n for positive integer n, but after continuation to n=0 the dominant contribution effectively comes from the saddle point with r= infinity . The perturbation expansion is found to be oscillatory at large orders and its behaviour is calculated.