Factoring the Laplacian to understand topological polymers

¹ University of Georgia - Athens, GA, USA
² Ochanomizu University - Tokyo, Japan
³ Colorado State University - Fort Collins, CO, USA
⁴ Kyoto University - Kyoto, Japan

Received: 18 March 2025
Accepted: 17 September 2025

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.