2022/05/18 by Jason Cantarella, Cantarella, Jason, Tetsuo Deguchi +5
Computer Science · Physics and Astronomy · #28A50 (secondary) #60D05 #60G50 #82D60 (primary) #Complex Network Analysis Techniques #Data Visualization and Analytics #FOS: Mathematics #FOS: Physical sciences #Probability (math.PR) #Statistical Mechanics (cond-mat.stat-mech) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2205.09049
openalex publication_date 2022/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝd can be written as a collection of d coordinate vectors, each in ℝn. 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.