Random graph embeddings with general edge potentials

TL;DR

This paper develops a theoretical framework for random embeddings of polymer networks with general potentials, revealing uncorrelated coordinate entries and providing methods to simplify complex graph distributions for easier computation.

Contribution

It introduces a uncorrelated coordinate property for polymer embeddings and a homology-based approach to simplify distributions of complex graph embeddings.

Findings

Entries from different coordinate vectors are uncorrelated.

A homology-inspired method to reduce complex graph distributions.

New formulas for edge covariances in phantom network theory.

Abstract

In this paper, we study random embeddings of polymer networks distributed according to any potential energy which can be expressed in terms of distances between pairs of monomers. This includes freely jointed chains, steric effects, Lennard-Jones potentials, bending energies, and other physically realistic models. A configuration of $n$ monomers in $\mathbb{R}^d$ can be written as a collection of $d$ coordinate vectors, each in $\mathbb{R}^n$. Our first main result is that entries from different coordinate vectors are uncorrelated, even when they are different coordinates of the same monomer. We predict that this property holds in realistic simulations and in actual polymer configurations (in the absence of an external field). Our second main contribution is a theorem explaining when and how a probability distribution on embeddings of a complicated graph may be pushed forward to a distribution on embeddings of a simpler graph to aid in computations. This construction is based on the idea of chain maps in homology theory. We use it to give a new formula for edge covariances in phantom network theory and to compute some expectations for a freely-jointed network.