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 $\mathrm{\Theta }$-point. In the $\mathrm{\Theta }$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, ${\gamma }_b^{\mathrm{\Theta }}$, to that of terminally-attached arches, ${\gamma }_{11}^{\mathrm{\Theta }},$ and to the correlation length exponent ${\nu }^{\mathrm{\Theta }}\mathrm{.}$ We find ${\gamma }_b^{\mathrm{\Theta }}={\gamma }_{11}^{\mathrm{\Theta }}+{\nu }^{\mathrm{\Theta }}\mathrm{.}$ In the case of the special transition, we find $\gamma_b^{\Theta}({\rm sp}) = \frac{1}{2}[\gamma_{11}^{\Theta}({\rm sp})+\gamma_{11}^{\Theta}]+\nu^{\Theta}.$ For general networks, the explicit expression of configurational exponents then naturally involve 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 case of ordinary, mixed and special surface transitions, and of the $\mathrm{\Theta }$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.