Branched Polymers and Hyperplane Arrangements

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 (Ann Math, 158:1019–1039, 2003), and Kenyon and Winkler (Am Math Mon, 116(7):612–628, 2009) to any central hyperplane arrangement $mathcal{A }$ A . The volume of the resulting configuration space of connected branched polymers associated to the hyperplane arrangement $mathcal{A }$ A is expressed through the value of the characteristic polynomial of $mathcal{A }$ 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 $mathcal{A }$ A at $-q$ ? 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.