Large order behavior for self-avoiding membranes Original Research Article

We derive the large order behavior of the perturbative expansion for the continuous model of tethered self-avoiding membranes. It is controlled by a classical configuration for an effective potential in bulk space, which is the analog of the Lipatov instanton, solution of a highly non-local equation. The nth order is shown to have factorial growth as (−cst)n (n!)(1−ϵ/D), where D is the “internal” dimension of the membrane and ϵ the engineering dimension of the coupling constant for self-avoidance. The instanton is calculated within a variational approximation, which is shown to become exact in the limit of large dimension d of bulk space. This is the starting point of a systematic 1/d expansion. As a consequence, the ϵ-expansion of self-avoidinng membranes has a factorial growth, like the ϵ-expansion of polymers and standard critical phenomena, suggesting Borel summability. Consequences for the applicability of the 2-loop calculations are examined.