Introduction to the theory of heaps of pieces with applications to statistical mechanics and quantum gravity
Seminar Room 1, Newton Institute
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 theoretical physics.
We state three basic lemmas 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 lemmas, once the translation of the problem into heaps methodology has been made.
Applications and interactions in statistical mechanics will be given with the now classical directed animal models and gas models with hard core interaction, such as Baxter's hard hexagons model; the appearance of some q-Bessel functions in two lattice models: the staircase polygons and the Solid-on-Solid model; and some 2D Lorentzian quantum gravity models introduced by Ambjorn, Loll, Di Francesco, Guitter and Kristjansen.
Finally, I shall present a heap bijective proof of a new identity of Bauer about loop-erased walks, in relation with recent work of Brydges and Abdesselam about loop ensembles and Mayer expansion.
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