## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 29-62 |

Number of pages | 34 |

Journal | Journal of Statistical Mechanics: Theory and Experiment |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2005 |

Externally published | Yes |

## Keywords

- Classical integrability
- Exact results
- Networks
- Random graphs
- Topology and combinatorics

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Statistics, Probability and Uncertainty