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 $Θ$-point. In the $Θ$-point case, generalising the good-solvent case, we relate the critical exponent for simple bridges, $γ_b^Θ$, to that of terminally-attached arches, $γ_{11}^Θ,$ and to the correlation length exponent $ν^Θ.$ We find $γ_b^Θ = γ_{11}^Θ+ν^Θ.$ In the case of the special transition, we find $γ_b^Θ({\rm sp}) = \frac{1}{2}[γ_{11}^Θ({\rm sp})+γ_{11}^Θ]+ν^Θ.$ 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 $Θ$-point. We provide compelling numerical evidence for some of these results both in two- and three-dimensions.

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