Affiliation: Universite Bordeaux 1
Email: viennot@labri.fr
Title Of Talk: Heaps of pieces in physics
Abstract: The notion of "heaps of pieces" has been introduced by the author in 1985, as a "geometrization" of the algebraic notion of commutation monoids defined by Cartier and Foata. The theory has been developed by the Bordeaux group of combinatorics, with strong interaction with statistical mechanics. Recently heaps of pieces has been used again by physicists for the resolution of some models in 2D Lorentzian quantum gravity (Di Francesco et al).
We begin by stating three basic lemma of the theory: an "inversion lemma" giving generating functions of heaps as the quotient of two alternating generating functions of "trivial" heaps, the "logarithmic lemma", and the "path lemma" saying that any path can be put in bijection with a heap. Many results and explicit formulae or identities in various papers scattered in the combinatorics and physics literature can be unified and viewed as consequence of these three basic lemma, once the translation of the problem into heaps methodology has been made.
The first interaction is with the now classical directed animal models and gas models with hard core interaction, such as Baxter's hard hexagons model. Combinatorial interpretation of the density of the gas is given (as power series in the variable fugacity), relating the model with directed animals (as given by Dhar,..). Very recently, the "multidirected" animal problem was solved (Bouquet-Melou, Rechnitzer) using heaps methodology.
The second topics is a unifying explanation of the appearance of some q-Bessel functions in two lattice models: the staircase polygons (or parallelogram polyominoes) (Bender, Delest, Fedou, ..) and the Solid-on-Solid model (Owczarek, Prellberg,..). It is pleasant to remark that by restricting the pieces of the heaps used in this unified interpretation, one get the Ramanujan continued fraction and also the Andrews interpretation of the reciprocal of Rogers-Ramanujan identities. The subject is also related to the construction of basis of the classical Temperley-Lieb algebra and the determination of its Hilbert polynomial.
Finally I will explain the recent use of heaps of pieces methodology for solving some Lorentzian quantum gravity models (Ambjorn, Loll , Di Francesco, Guitter, .. and some joint work with W. James). In particular we relate the appearance of Bessel functions in the continuum limit of the model to the above solutions of staircase polygons and Solid-on-Solid models. The multidirected animals mentioned above play a role in this quantum gravity story.
Last update made Wed Mar 26 14:55:31 EST 2003.
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