Geodesic Distance in Planar Graphs: An Integrable Approach |
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Authors: | Email author" target="_blank">P?Di?FrancescoEmail author |
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Institution: | (1) Service de Physique Théorique, CEA/DSM/SPhT, Unité de recherche associée au CNRS, CEA/Saclay, 91191 Gif sur Yvette Cedex, France |
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Abstract: | We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track
of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations
on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the
KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked
points at large geodesic distances, and obtain integrable differential equations for the corresponding scaling functions.
This provides a continuum formulation of two-dimensional quantum gravity, in terms of the geodesic distance.
2000 Mathematics Subject Classification: Primary—05C30; Secondary—05A15, 05C05, 05C12, 68R05 |
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Keywords: | planar maps integrable systems geodesic distance solitons |
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