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### Abstract

We study a configuration model on bipartite planar maps in which, given $n$ even integers, one samples a planar map with $n$ faces uniformly at random with these face degrees. We prove that when suitably rescaled, such maps always admit nontrivial subsequential limits as $n\to \infty $ in the Gromov–Hausdorff–Prokhorov topology. Further, we show that they converge in distribution towards the celebrated Brownian sphere, and more generally a Brownian disk for maps with a boundary, if and only if there is no inner face with a macroscopic degree, or, if the perimeter is too big, the maps degenerate and converge to the Brownian tree. By first sampling the degrees at random with an appropriate distribution, this model recovers that of size-conditioned Boltzmann maps associated with critical weights in the domain of attraction of a stable law with index $\alpha \in [1,2]$. The Brownian tree and disks then appear respectively in the case $\alpha =1$ and $\alpha =2$, whereas in the case $\alpha \in (1,2)$ our results partially recover previous known ones. Our proofs rely on known bijections with labelled plane trees, which are similarly sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some results on the geometry of such trees, such as a convergence to the Brownian tree but only in the weaker sense of subtrees spanned by random vertices, which are of independent interest.

### References

[AB12] Tail bounds for the height and width of a random tree with a given degree sequence, Random Struct. Algorithms, Volume 41 (2012) no. 2, pp. 253-261 | DOI | MR | Zbl

[ABA17] The scaling limit of random simple triangulations and random simple quadrangulations, Ann. Probab., Volume 45 (2017) no. 5, pp. 2767-2825 | DOI | Zbl

[ABA21] Convergence of non-bipartite maps via symmetrization of labeled trees, Ann. Henri Lebesgue, Volume 4 (2021), pp. 653-683 | DOI | MR | Zbl

[Abr16] Rescaled bipartite planar maps converge to the Brownian map, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 52 (2016) no. 2, pp. 575-595 | DOI | MR | Zbl

[Ald85] Exchangeability and related topics, École d’été de probabilités de Saint-Flour, XIII—1983 (Lecture Notes in Mathematics), Volume 1117, Springer, 1985, pp. 1-198 | DOI | MR | Zbl

[Ald93] The continuum random tree. III, Ann. Probab., Volume 21 (1993) no. 1, pp. 248-289 | DOI | MR | Zbl

[AMP04] The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity, Probab. Theory Relat. Fields, Volume 129 (2004) no. 2, pp. 182-218 | DOI | MR | Zbl

[BCM19] A Boltzmann approach to percolation on random triangulations, Can. J. Math., Volume 71 (2019) no. 1, pp. 1-43 | DOI | MR | Zbl

[BCP03] Path transformations of first passage bridges, Electron. Commun. Probab., Volume 8 (2003), pp. 155-166 | DOI | MR | Zbl

[BDFG04] Planar maps as labeled mobiles, Electron. J. Comb., Volume 11 (2004) no. 1, 69 | MR | Zbl

[Ber19] Notes on random walks in the Cauchy domain of attraction, Probab. Theory Relat. Fields, Volume 175 (2019) no. 1-2, pp. 1-44 | DOI | MR | Zbl

[Bet10] Scaling limits for random quadrangulations of positive genus, Electron. J. Probab., Volume 15 (2010), 52, pp. 1594-1644 | DOI | MR | Zbl

[Bet15] Scaling limit of random planar quadrangulations with a boundary, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 2, pp. 432-477 | DOI | Numdam | MR | Zbl

[BJM14] The scaling limit of uniform random plane maps, via the Ambjørn–Budd bijection, Electron. J. Probab., Volume 19 (2014), 74 | DOI | MR | Zbl

[BM14] Asymptotics of trees with a prescribed degree sequence and applications, Random Struct. Algorithms, Volume 44 (2014) no. 3, pp. 290-316 | DOI | MR | Zbl

[BM17] Compact Brownian surfaces I: Brownian disks, Probab. Theory Relat. Fields, Volume 167 (2017) no. 3-4, pp. 555-614 | DOI | MR | Zbl

[CLGM13] The Brownian cactus I. Scaling limits of discrete cactuses, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 49 (2013) no. 2, pp. 340-373 | DOI | Numdam | MR | Zbl

[Fel71] An introduction to probability theory and its applications. Vol. II., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, 1971 | MR | Zbl

[Hoe63] Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc., Volume 58 (1963), pp. 13-30 | DOI | MR | Zbl

[JS15] Scaling limits of random planar maps with a unique large face, Ann. Probab., Volume 43 (2015) no. 3, pp. 1045-1081 | DOI | MR | Zbl

[Kor12] Invariance principles for Galton–Watson trees conditioned on the number of leaves, Stochastic Processes Appl., Volume 122 (2012) no. 9, pp. 3126-3172 | DOI | MR | Zbl

[Kor15] Limit theorems for conditioned non-generic Galton–Watson trees, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 51 (2015) no. 2, pp. 489-511 | DOI | Numdam | MR | Zbl

[Kor17] Sub-exponential tail bounds for conditioned stable Bienaymé–Galton–Watson trees, Probab. Theory Relat. Fields, Volume 168 (2017) no. 1-2, pp. 1-40 | DOI | MR | Zbl

[KR19] Condensation in critical Cauchy Bienaymé–Galton–Watson trees, Ann. Appl. Probab., Volume 29 (2019) no. 3, pp. 1837-1877 | DOI | Zbl

[KR20] The boundary of random planar maps via looptrees, Ann. Fac. Sci. Toulouse, Math., Volume 29 (2020) no. 2, pp. 391-430 | DOI | MR | Zbl

[Lei19] Scaling limit of random forests with prescribed degree sequences, Bernoulli, Volume 25 (2019) no. 4A, pp. 2409-2438 | DOI | MR | Zbl

[LG05] Random trees and applications, Probab. Surv., Volume 2 (2005), pp. 245-311 | DOI | MR | Zbl

[LG07] The topological structure of scaling limits of large planar maps, Invent. Math., Volume 169 (2007) no. 3, pp. 621-670 | DOI | MR | Zbl

[LG13] Uniqueness and universality of the Brownian map, Ann. Probab., Volume 41 (2013) no. 4, pp. 2880-2960 | DOI | MR | Zbl

[LGM11] Scaling limits of random planar maps with large faces, Ann. Probab., Volume 39 (2011) no. 1, pp. 1-69 | DOI | MR | Zbl

[LGP08] Scaling limits of bipartite planar maps are homeomorphic to the 2-sphere, Geom. Funct. Anal., Volume 18 (2008) no. 3, pp. 893-918 | DOI | MR | Zbl

[Mar18a] On scaling limits of planar maps with stable face-degrees, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 15 (2018) no. 2, pp. 1089-1123 | DOI | MR | Zbl

[Mar18b] Scaling limits of random bipartite planar maps with a prescribed degree sequence, Random Struct. Algorithms, Volume 53 (2018) no. 3, pp. 448-503 | DOI | MR | Zbl

[McD98] Concentration, Probabilistic methods for algorithmic discrete mathematics (Algorithms and Combinatorics), Volume 16, Springer, 1998, pp. 195-248 | DOI | MR | Zbl

[Mie08] On the sphericity of scaling limits of random planar quadrangulations, Electron. Commun. Probab., Volume 13 (2008), pp. 248-257 | DOI | MR | Zbl

[Mie09] Tessellations of random maps of arbitrary genus, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 725-781 | DOI | Numdam | MR | Zbl

[Mie13] The Brownian map is the scaling limit of uniform random plane quadrangulations, Acta Math., Volume 210 (2013) no. 2, pp. 319-401 | DOI | MR | Zbl

[MM03] The depth first processes of Galton–Watson trees converge to the same Brownian excursion, Ann. Probab., Volume 31 (2003) no. 3, pp. 1655-1678 | DOI | MR | Zbl

[MM07] Invariance principles for random bipartite planar maps, Ann. Probab., Volume 35 (2007) no. 5, pp. 1642-1705 | DOI | MR | Zbl

[Pit06] Combinatorial stochastic processes. École d’Été de Probabilités de Saint-Flour XXXII – 2002., Lecture Notes in Mathematics, 1875, Springer, 2006 | DOI | Zbl

[Thé20] Vertices with fixed outdegrees in large Galton–Watson trees, Electron. J. Probab., Volume 25 (2020), 64 | DOI | MR | Zbl