Factoring the Laplacian to understand topological polymers

TL;DR

By factoring the graph Laplacian as $L=BB^T$, the paper reframes loop constraints of topological polymers in edge space and shows that edge displacements form a centered Gaussian on $ ext{col } B^T$ with covariance $(1/d)L^+$. It presents efficient spectral methods for sampling embeddings, derives exact expressions for correlation functions, and generalizes the framework to non-Gaussian distributions, enabling analysis of radius of gyration and other observables. The approach provides a flexible, computational toolkit for phantom elastic networks with complex topology, with potential extensions to self-avoidance and more realistic polymer models.

Abstract

A ring polymer is a random walk whose steps obey a single linear condition; their sum vanishes. Factoring the graph Laplacian into the product of the incidence matrix and its transpose allows us to show that for a more complicated network, the steps must lie in a linear subspace determined by the graph topology. This provides a useful new perspective on the James--Guth theory of phantom elastic networks. In particular, we formulate phantom networks which are free from the constraints of fixed crosslinks. For a given network the solution of the loop constraints makes the partition function finite-valued in the path integral formulation without applying any external forces or fixing any monomer positions. The resulting probability distribution on edge displacements is rotationally invariant, which is practically quite useful for generating unbiased random samples of edge displacements and monomer positions. Furthermore, one can exactly calculate many physical quantities such as correlation functions with respect to this distribution. Finally, this reformulation lends itself well to the case of non-Gaussian distributions. We illustrate this by computing the expected radius of gyration of a ring polymer in a wide variety of models.

Figures (3)

• Figure 1: A particular graph embedding in $\mathbb{R}^2$, along with the components of its vertex vector $x \in (\mathbb{R}^{2})^4$ and edge vector $w \in (\mathbb{R}^{2})^4$.
• Figure 2: In these pictures, we construct a Gaussian ring polymer $x \in \mathbb{R}^3$ with $1000$ edges by sampling $y^1, y^2, y^3 \in \mathbb{R}^{1000}$ from $\mathcal{N}\left(0,\frac{1}{3}I_{1000}\right)$ and computing $x^k_p = (L^+)^{1/2}_p y^k$ for various low-rank approximations $(L^+)^{1/2}_p$ of $(L^+)^{1/2}$. We can see that the low-rank approximations model the polymer rather well.
• Figure 3: In these pictures, we construct a Gaussian random embedding $x \in \mathbb{R}^3$ of a 1499 vertex $\theta$-curve where each edge has been subdivided into 500 pieces by sampling $y^1, y^2, y^3 \in \mathbb{R}^{1499}$ from $\mathcal{N}\left(0,\frac{1}{3}I_{1499}\right)$ and computing $x^k_p = (L^+)^{1/2}_p y^k$ for various low-rank approximations $(L^+)^{1/2}_p$ of $(L^+)^{1/2}$. Again, the low-rank approximations model the polymer rather well.

Theorems & Definitions (20)

• Definition 1
• Theorem 2
• proof
• Theorem 3
• Definition 4
• Theorem 5
• proof
• Corollary 6
• Corollary 7
• proof