Transfer matrix for spanning trees, webs and colored forests

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.

29 pages, 7 figures

References in corpus (8)

• A Local Logarithmic Conformal Field Theory
• Spanning forests and the vector bundle Laplacian
• Logarithmic Conformal Field Theory and Boundary Effects in the Dimer Model
• A c=-2 boundary changing operator for the Abelian sandpile
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Dimer statistics on the Möbius strip and the Klein bottle
• Vacancy localization in the square dimer model
• Jamming probabilities for a vacancy in the dimer model

Cited by in corpus (4)

• Integrability and conformal data of the dimer model
• Dimer representations of the Temperley-Lieb algebra
• Multipoint correlators in the Abelian sandpile model
• Bimodule structure of the mixed tensor product over $U_{q}s\ell (2|1)$ and quantum walled Brauer algebra