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

中文翻译：

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受限聚合物网络的统计力学

我们展示了如何将任意拓扑的 d 维聚合物网络的临界行为理论推广到由超平面限制的网络的情况。这特别包括桥构型的单个聚合物链的情况。我们进一步定义了多桥网络，其中多个顶点处于本地桥配置中。我们考虑所有普通的、混合的和特殊的表面过渡，以及由自回避游走、相互回避游走或三临界 $$\Theta $$ Θ 点构成的聚合物链的所有情况。在 $$\Theta $$ Θ 点的情况下，概括了良好溶剂的情况，我们将简单桥的临界指数 $$\gamma _b^{\Theta }$$ γ b Θ 与终端的临界指数-附加拱形，$$\gamma _{11}^{\Theta },$$ γ 11 Θ ，以及相关长度指数 $$\nu ^{\Theta }.$$ ν Θ 。我们发现 $$\gamma _b^{\Theta } =\gamma _{11}^{\Theta }+\nu ^{\Theta }$$ γ b Θ = γ 11 Θ + ν Θ 。在特殊跃迁的情况下，我们发现 $$\gamma _b^{\Theta }(\mathrm{sp}) =\frac{1}{2}[\gamma _{11}^{\Theta }(\ mathrm{sp})+\gamma _{11}^{\Theta }]+\nu ^{\Theta }$$ γ b Θ ( sp ) = 1 2 [ γ 11 Θ ( sp ) + γ 11 Θ ] + νθ。对于一般网络，配置指数的显式表达自然会涉及多个随机路径的体指数和表面指数。在二维中，我们使用刘维尔量子引力 (LQG) 中的 Schramm-Loewner Evolution (SLE) 以及欧几里得和 LQG 标度维之间的所谓 KPZ 关系，从统一的角度描述它们的欧几里得指数。这是在普通、混合和特殊表面过渡以及 $$\Theta $$ Θ 点的情况下完成的。