We study numerically the continuum limit corresponding to the nontrivial fixed point of Dyson’s hierarchical model. We discuss the possibility of using the critical amplitudes as input parameters. We determine numerically the leading and subleading critical amplitudes of the zero-momentum connected $2l$-point functions in the symmetric phase up to the 20-point function for randomly chosen local measures. Using these amplitudes, we construct quantities which are expected to be universal in the limit where very small log-periodic corrections are neglected: the $U^{\mathrm{}\left(2l\right)\mathrm{}\star }$ (proportional to the connected $2l$-point functions) and the $r_{2l}$ (proportional to one-particle irreducible functions). We show that these quantities are independent of the local measure with at least five significant digits. We provide clear evidence for the asymptotic behavior $U^{\mathrm{}\left(2l\right)\mathrm{}\star }\propto \mathrm{}\left(2l\right)$ and reasonable evidence for $r_{2l}\propto \mathrm{}\left(2l\right).\mathrm{}$ These results signal a finite radius of convergence for the generating functions. We provide numerical evidence for a linear growth for universal ratios of subleading amplitudes. We compare our $r_{2l}$ with existing estimates for other models.

• Received 21 January 2004

DOI:https://doi.org/10.1103/PhysRevD.69.125016