Xavier Viennot - University of Bordeaux
Multidirected animals, heaps of dimers and Lorentzian triangulations
Bousquet-Melou and Rechnitzer have introduced the class of multidirected animals, as an extension of the classical 2D directed animals. They gave an explicit expression for their generating function. In the case of directed animals, the corresponding generating function is algebraic and various (easy) "combinatorial explanations" (bijective proof) have been given, in particular using the so-called model of heaps of dimers. Although the generating function for multidirected animals is not algebraic, even worst not D-finite, we give a bijective proof of Bousquet-Melou and Rechnitzer's formula, introducing the "Nordic decomposition" of a connected heaps of dimers.
One possible interest of this bijective proof is in relation with 2D Lorentzian quantum gravity. There exist correspondences between the 2D Lorentzian triangulations introduced and studied by Ambjorn, Loll, Di Francesco, Guitter, Kristjansen, .. , connected heaps of dimers and multidirected animals.