  We compute the normalisation factor for the large order asymptotics of
perturbation theory for the self-avoiding manifold (SAM) model describing
flexible tethered (D-dimensional) membranes in d-dimensional space, and the
epsilon-expansion for this problem. For that purpose, we develop the methods
inspired from instanton calculus, that we introduced in a previous publication
(Nucl. Phys. B 534 (1998) 555), and we compute the functional determinant of
the fluctuations around the instanton configuration. This determinant has UV
divergences and we show that the renormalized action used to make perturbation
theory finite also renders the contribution of the instanton UV-finite. To
compute this determinant, we develop a systematic large-d expansion. For the
renormalized theory, we point out problems in the interplay between the limits
epsilon->0 and d->infinity, as well as IR divergences when epsilon= 0. We show
that many cancellations between IR divergences occur, and argue that the
remaining IR-singular term is associated to amenable non-analytic contributions
in the large-d limit when epsilon= 0. The consistency with the standard
instanton-calculus results for the self-avoiding walk is checked for D = 1.
