Transfer matrix for spanning trees, webs and colored forests

TL;DR

This paper extends Lieb's transfer matrix formalism to analyze arrow configurations, spanning trees, and forests on a cylinder, introducing a new matrix that reveals Jordan cells and sector decomposition.

Contribution

It generalizes the transfer matrix approach to include colored forests and identifies Jordan cells in the extended matrix, advancing the understanding of these combinatorial structures.

Findings

Partition functions for arrow configurations on a cylinder are computed.

The extended transfer matrix reveals Jordan cells in the spectrum.

A sector decomposition based on non-contractible loops is established.

Abstract

We use the transfer matrix formalism for dimers proposed by Lieb, and generalize it to address the corresponding problem for arrow configurations (or trees) associated to dimer configurations through Temperley's correspondence. On a cylinder, the arrow configurations can be partitioned into sectors according to the number of non-contractible loops they contain. We show how Lieb's transfer matrix can be adapted in order to disentangle the various sectors and to compute the corresponding partition functions. In order to address the issue of Jordan cells, we introduce a new, extended transfer matrix, which not only keeps track of the positions of the dimers, but also propagates colors along the branches of the associated trees. We argue that this new matrix contains Jordan cells.