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Séminaire Lotharingien de Combinatoire, 78B.79 (2017), 12 pp.

# Gillaume Chapuy

# On Tessellations of Random Maps and the *t*_{g}-Recurrence

**Abstract.**
The number of *n*-edge embedded graphs (rooted maps) on the *g*-torus
grows as *t*_{g}*n*^{5(g-1)/2}12^{n} when *n* tends to infinity. The
constants *t*_{g} can be computed via the non-linear
"*t*_{g}-recurrence",
strongly related to the KP hierarchy and the
double scaling limit of the one-matrix model. The combinatorial
meaning of this simple recurrence is still mysterious, and the purpose
of this work is to point out an interpretation via random maps on
surfaces. Namely, we show that the
*t*_{g}-recurrence is equivalent,
via combinatorial bijections, to the fact that
**E***X*_{g}^{2}
= 1/3 for any *g* >= 0, where
*X*_{g},1-*X*_{g} are
the masses of the nearest-neighbour cells surrounding two randomly
chosen points in a Brownian map of genus *g*. This raises the question
(that we leave open) of giving an independent probabilistic or
combinatorial derivation of this second moment, which would lead to a
fully concrete proof of the *t*_{g}-recurrence. In fact, we conjecture
that for any *g* >= 0 and *k* >= 2, the masses of the *k*
nearest-neighbour cells induced by *k* uniform points in the genus *g*
Brownian map form a uniform *k*-division of the unit interval. We
leave this question open even for (*g*,*k*)=(0,2).

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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