Branched polymers and hyperplane arrangements

TL;DR

This paper extends the concept of branched polymers to arbitrary hyperplane arrangements, linking their configuration space volume to the characteristic polynomial and connecting topological properties to algebraic structures.

Contribution

It generalizes the construction of branched polymers to any hyperplane arrangement and relates their volume to the characteristic polynomial, also exploring cohomology and algebraic structures.

Findings

Volume of connected branched polymers equals the characteristic polynomial at 0.

Introduces q-volume for non-connected polymers via the characteristic polynomial at -q.

Shows the cohomology ring is isomorphic to the Orlik-Solomon algebra.

Abstract

We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie, and Kenyon and Winkler to any hyperplane arrangement A. The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement A is expressed through the value of the characteristic polynomial of A at 0. We give a more general definition of the space of branched polymers, where we do not require connectivity, and introduce the notion of q-volume for it, which is expressed through the value of the characteristic polynomial of A at -q. Finally, we relate the volume of the space of branched polymers to broken circuits and show that the cohomology ring of the space of branched polymers is isomorphic to the Orlik-Solomon algebra.