Combinatorial aspects of the Painlevé-I recurrence for maps on surfaces, and Voronoï tessellations of Brownian maps.

12/08/2015 - 15:30
12/08/2015 - 16:30
Guillaume Chapuy (CNRS, CRM et Université Paris Diderot)
CRM, UdeM, Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle 4336

The number m_g(n) of maps (fat graphs) with n edges on the g-torus satisfies an asymptotic of the form m_g(n) ~ t_g n^{5(g-1)/2} 12^n when g is fixed and n tends to infinity. The universal constants t_g can be computed thanks to a non-linear recurrence formula of Painlevé-I type (related to the "double-scaling limit" of the one-matrix model).

Although the mathematical integrability underlying this fact is well understood, this simple "topological" recurrence appeals for a direct combinatorial interpretation which is, so far, not known. The purpose of the talk is to relate this question to properties of continuum random surfaces of genus g (a.k.a. "Brownian maps", conjecturally linked to "Liouville quantum gravity"). Namely, we give a simple recursive slicing procedure of the surfaces that gives rise to a non-linear recurrence for the numbers t_g. This recurrence features unknown constants that are naturally interpreted as observables of nearest-neighbour "Voronoï tessellations" around random vertices in Brownian surfaces. By comparing with the Painlevé-I equation, we identify these moments, which are unexpectedly simple and suggest that our (my) understanding of high-genus random surfaces is still very incomplete.

Last edited by on Wed, 12/02/2015 - 11:12