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

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.

Field theory of the RNA freezing transition

“…It depends both on f R and on g R . For the disorder-free model g R = 0, one recovers the renormalized theory for the model of a random walk pinned by an impurity [29,30,54,55,56]. The renormalized action can be written as a bare action S R [r R ] = S B [r B ], with the same bare fields and bare couplings r B and g B as before, but with a bare force f B given by…”

“…For the disorder-free model g R = 0, one recovers the renormalized theory for the model of a random walk pinned by an impurity [29,30,54,55,56].…”

Dynamical selection of critical exponents

“…We also remark that given u c , the amplitude R(0) is not fixed by the flow equation (14), even though it is fixed for the solutions of Eq. (22). Indeed, changing R(u) → κR(u), this can be absorbed into a change of → κ .…”

“…(iii) Another possibility is to consider short-ranged potentials V(u) δ(u) from the start. This leads to a renormalizable field theory, pioneered by David, Duplantier and Guitter [16,17], and further studied by several authors [18][19][20][21][22]. It is this approach that has been successful to tackle the renormalization of self-avoiding manifolds [23][24][25][26][27][28][29][30][31][32].…”