Statistical Mechanics of Confined Polymer Networks

We show how the theory of the critical behaviour of d-dimensional polymer networks of arbitrary topology can be generalized to the case of networks confined by hyperplanes. This in particular encompasses the case of a single polymer chain in a bridge configuration. We further define multi-bridge networks, where several vertices are in local bridge configurations. We consider all cases of ordinary, mixed and special surface transitions, and polymer chains made of self-avoiding walks, or of mutually-avoiding walks, or at the tricritical $$\Theta $$ -point. In the $$\Theta $$ -point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $$\gamma _b^{\Theta }$$ , to that of terminally-attached arches, $$\gamma _{11}^{\Theta },$$ and to the correlation length exponent $$\nu ^{\Theta }.$$ We find $$\gamma _b^{\Theta } =\gamma _{11}^{\Theta }+\nu ^{\Theta }$$ . In the case of the special transition, we find $$\gamma _b^{\Theta }(\mathrm{sp}) =\frac{1}{2}[\gamma _{11}^{\Theta }(\mathrm{sp})+\gamma _{11}^{\Theta }]+\nu ^{\Theta }$$ . For general networks, the explicit expression of configurational exponents then naturally involves bulk and surface exponents for multiple random paths. In two-dimensions, we describe their Euclidean exponents from a unified perspective, using Schramm–Loewner Evolution (SLE) in Liouville quantum gravity (LQG), and the so-called KPZ relation between Euclidean and LQG scaling dimensions. This is done in the cases of ordinary, mixed and special surface transitions, and of the $$\Theta $$ -point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.