Scaling behavior of tethered crumpled manifolds with inner dimension close to D=2: Resumming the perturbation theory

TL;DR

This paper develops an exact summation of perturbation theory for tethered manifolds near D=2, providing analytic solutions and connecting to polymer models, with implications for understanding self-avoiding membranes.

Contribution

It introduces a method to sum perturbation expansions exactly in the limit D->2, offering new analytic insights into tethered manifolds and their strong-coupling behavior.

Findings

Exact summation of perturbation series in D->2 limit

Analytic solutions for strong-coupling regime

Proposed Monte-Carlo tests for theoretical predictions

Abstract

The field theory of self-avoiding tethered membranes still poses major challenges. In this article, we report progress on the toy-model of a manifold repelled by a single point. Our approach allows to sum the perturbation expansion in the strength g_0 of the interaction exactly in the limit of internal dimension D -> 2, yielding an analytic solution for the strong-coupling limit. This analytic solution is the starting point for an expansion in 2-D, which aims at connecting to the well studied case of polymers (D=1). We give results to fourth order in 2-D, where the dependence on g_0 is again summed exactly. As an application, we discuss plaquette density functions, and propose a Monte-Carlo experiment to test our results. These methods should also allow to shed light on the more complex problem of self-avoiding manifolds.