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On tessellations of random maps and the \(t_g\)-recurrence

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Abstract

We study the masses of the two cells in a Voronoï tessellation of the Brownian surface of genus \(g\ge 0\) centered on two uniform random points. Making use of classical bijections and asymptotic estimates for maps of fixed genus, we relate the second moment of these random variables to the Painlevé-I equation satisfied by the double scaling limit of the one-matrix model, or equivalently to the “\(t_g\)-recurrence” satisfied by the constants \(t_g\) driving the asymptotic number of maps of genus \(g\ge 0\). This raises the question of giving an independent probabilistic or combinatorial derivation of this second moment, which would then lead to new proof of the \(t_g\)-recurrence. More generally we conjecture that for any \(g\ge 0\) and \(k\ge 2\), the masses of the cells in a Voronoï tessellation of the genus-g Brownian surface by k uniform points follows a Dirichlet\((1,1,\ldots ,1)\) distribution.

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Notes

  1. If \(U_1,U_2\) are two independent uniforms on [0, 1] and \(I_1, I_2, I_3\) are the lengths of the three intervals they define, then \(\mathbf {E} (I_1I_2I_3)\) is the probability that five independent uniforms \(U_1,U_2,V_1,V_2,V_3\) are ordered as \(V_1<U_{1}\wedge U_2<V_2<U_1\vee U_{2}<V_3\), which is clearly equal to \(\frac{2}{5!}=\tfrac{1}{60}\).

  2. After a first version of this paper was made public, Emmanuel Guitter was able to apply this idea, and checked by a “semi-rigourous” calculation that for \((g,k)=(0,2)\) the law is indeed uniform [18]. Guitter’s remarkable calculation is computer assisted and very heavy, moreover fully justifying all of the needed approximations seems technically difficult—yet it strongly supports our conjecture. This method seems unfit to apply to general values of k (or g), and probably too heavy to apply to any other value of (g, k).

  3. We thank a referee for this remark.

  4. Strictly speaking, these notions are defined only for dominant maps in that reference. Here it will be convenient for the presentation of the decompositions to extend them to general 3-nodes—but this is not a fundamental need, since all quantities involved in our discussion will be led by dominant maps at the first order.

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Authors and Affiliations

  1. IRIF, UMR CNRS 8243, Université Paris-Diderot, Paris, France

    Guillaume Chapuy

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  1. Guillaume Chapuy
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Correspondence to Guillaume Chapuy.

Additional information

Support from the European Research Council, Grant ERC-2016-STG 716083 “CombiTop”, from Agence Nationale de la Recherche, Grant Number ANR 12-JS02-001-01 “Cartaplus”, and from the City of Paris, Grant “Émergences Paris 2013, Combinatoire à Paris”. Part of this research was done while I was affiliated with the CRM, UMI CNRS 3457, Université de Montréal, Canada.

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Chapuy, G. On tessellations of random maps and the \(t_g\)-recurrence. Probab. Theory Relat. Fields 174, 477–500 (2019). https://doi.org/10.1007/s00440-018-0865-6

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  • DOI: https://doi.org/10.1007/s00440-018-0865-6

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