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Compact Brownian surfaces I: Brownian disks

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Abstract

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family \((\mathrm {BD}_L,\, 0<L<\infty )\) of random metric spaces homeomorphic to the closed unit disk of \(\mathbb {R}^2\), the space \(\mathrm {BD}_L\) being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where \(L=0\). Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.

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Notes

  1. By convention, the vertex map \(\circ \) consisting of no edges and only one vertex, “bounding” a face of degree 0, is considered as an element of \(\mathbf {B}\), so that \(\mathbf {B}_0=\{\circ \}\). It will only appear incidentally in the analysis.

  2. We could also consider other ways to measure the size of a map \(\mathbf {m}\), e.g. considering combinations of the form \(x_\mathbf {V}|\mathbf {V}(\mathbf {m})|+x_\mathbf {E}|\mathbf {E}(\mathbf {m})|+x_\mathbf {F}|\mathbf {F}(\mathbf {m})|\) for some \(x_\mathbf {V}\), \(x_\mathbf {E}\), \(x_\mathbf {F}\ge 0\) with sum 1 as is done for instance in [42] (in fact, due to the Euler formula, there is really only one degree of freedom rather than two). We will not address this here but we expect our results to hold in this context as well.

  3. The very recent preprints [24, 25] seem however to provide a decisive step towards the solution of this problem. There, the authors consider the similar problem of gluing two Brownian half-planes, a non-compact version of the Brownian disk introduced in [6] and [26], along their boundaries.

  4. We will use notation like \(Q_n\), \(C_n\), \(\ell _n\), \(D_n\), D with a different meaning from the preceding section in order to keep exposition lighter.

  5. This is of course an abuse of notation since (X, Z) previously denoted the canonical process, however we did not want to introduce a further specific notation at this point.

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Acknowledgements

This work is partly supported by ANR-14-CE25-0014 (GRAAL) and ANR-15-CE40-0013 (Liouville). We also acknowledge partial support from the Isaac Newton Institute for Mathematical Sciences where part of this work was conducted, and where G.M. benefited from a Rothschild Visiting Professor position during January 2015. We thank Erich Baur, Timothy Budd, Guillaume Chapuy, Nicolas Curien, Igor Kortchemski, Jean-François Le Gall, Jason Miller, Gourab Ray and Scott Sheffield, for useful remarks and conversations during the elaboration of this work. Thanks also to the very thorough reading of two anonymous referees, whose comments were greatly appreciated.

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  1. CNRS & Laboratoire d’Informatique de l’École Polytechnique, Palaiseau, France

    Jérémie Bettinelli

  2. ENS de Lyon & Institut Universitaire de France, Lyon, France

    Grégory Miermont

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  1. Jérémie Bettinelli
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  2. Grégory Miermont
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Correspondence to Grégory Miermont.

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2014 Wolfgang Doeblin Prize Article.

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Bettinelli, J., Miermont, G. Compact Brownian surfaces I: Brownian disks. Probab. Theory Relat. Fields 167, 555–614 (2017). https://doi.org/10.1007/s00440-016-0752-y

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