Integrability of graph combinatorics via random walks and heaps of dimers

Details Integrability of graph combinatorics via random walks and heaps of dimers Integrability of graph combinatorics via random walks and heaps of dimers

2005-06-27

arxiv.org/abs/math/0506542v1

J. Stat. Mech. (2005) P09001

Mathematics

Combinatorics

38 pages, 15 figures, uses epsf, lanlmac and hyperbasics

Scientific paper

10.1088/1742-5468/2005/09/P09001

We investigate the integrability of the discrete non-linear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for random walks. We construct explicitly an infinite set of conserved quantities for this equation, also involving suitable combinations of random walk generating functions. The proof of their conservation, i.e. their eventual independence on the geodesic distance, relies on the connection between random walks and heaps of dimers. The values of the conserved quantities are identified with generating functions for graphs with fixed numbers of external legs. Alternative equivalent choices for the set of conserved quantities are also discussed and some applications are presented.

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