## Geodesic stars in random geometry.(English)Zbl 1526.60016

A geodesic in a metric space $$(E, d)$$ is a continuous path $$(\gamma(t))_{t\in[0,\delta]}$$, where $$\delta > 0$$, such that $$d(\gamma(s), \gamma(t)) = |s - t|$$ for every $$s, t \in [0, \delta]$$. For every $$t \in(0, \delta)$$, $$\gamma(t)$$ is an interior point of the geodesic (whereas $$\gamma(0)$$ and $$\gamma(\delta)$$ are its endpoints). If $$m\ge 1$$ is an integer, then a point $$x$$ is a geodesic star with $$m$$ arms (in short, an $$m$$-geodesic star) if there exist $$\delta > 0$$ and $$m$$ geodesics $$(\gamma_1(t))_{t\in[0,\delta]}, \dots, (\gamma_m(t))_{t\in[0,\delta]}$$ such that $$\gamma_1(0) = \gamma_2(0) = \dots = \gamma_m(0) = x$$ and the sets $$\{\gamma_j(t): t \in (0, \delta]\}$$, for $$j \in \{1, \dots, m\}$$, are disjoint. If $$(E, d)$$ is a geodesic space, any pair of distinct points is connected by a (possibly not unique) geodesic, and it is then immediate that every point is a $$1$$-geodesic star.
The main result of the paper is the following statement.
Theorem 1. Let $$(m_{\infty}, D)$$ denote the Brownian sphere. For every integer $$m\in \{1, 2, 3, 4\}$$, let $${\mathcal{E}}_m$$ be the set of all $$m$$-geodesic stars in $$(m_\infty, D)$$. Then, the Hausdorff dimension of $${\mathcal{E}}_m$$ is a.s. equal to $$5-m$$.
The upper bound $$\dim({\mathcal{E}}_m) \leq 5 - m$$ has been obtained in [J. Miller and W. Qian, “Geodesics in the Brownian map: Strong confluence and geometric structure”, Preprint, arXiv:2008.02242]. So the contribution of the present work is to prove the corresponding lower bound. Note that $$m$$-geodesic stars in the Brownian sphere were first discussed in [G. Miermont, Acta Math. 210, No. 2, 319–401 (2013; Zbl 1278.60124)]. The Brownian sphere is a geodesic space, and thus $${\mathcal{E}}_1 = m_{\infty}$$ so that, in the case $$m = 1$$, the result follows from the known fact [J.-F. Le Gall, Invent. Math. 169, No. 3, 621–670 (2007; Zbl 1132.60013)] that $$\dim(m_{\infty}) = 4$$. Any interior point of a geodesic is a $$2$$-geodesic star, and, therefore, $${\mathcal{E}}_2$$ contains the set of all interior points of all geodesics. However, the authors of [Miller and Qian, loc. cit.] proved that the Hausdorff dimension of the latter set is 1 (it is obviously greater than or equal to 1), thus confirming a conjecture from [O. Angel et al., Ann. Probab. 45, No. 5, 3451–3479 (2017; Zbl 1407.60018)]. Since $$\dim({\mathcal{E}}_2) =3$$, this implies, that typical $$2$$-geodesic stars are not interior points of geodesics.
Open Problem. Prove or disprove the existence of $$5$$-geodesic stars in the Brownian map.
The paper is organized as follows. Section 2 is devoted to a number of preliminaries, including the Brownian snake construction of the Brownian sphere as a measure metric space with two distinguished points denoted by $$x_*$$ and $$x_0$$, and a discussion of the symmetry properties of the Brownian sphere, which, roughly speaking, say that $$x_*$$ and $$x_0$$ play the same role as two points chosen independently according to the (normalized) volume measure. Theorem 8 of Section 3 shows that the hull of radius $$r>0$$ centered at $$x_*$$ (relative to $$x_0$$) is independent of its complement conditionally on its boundary size, and the complement itself is a Brownian disk; this is, in fact, an analog of a result proved in [J.-F. Le Gall and A. Riera, Probab. Theory Relat. Fields 181, No. 1–3, 571–645 (2021; Zbl 1480.60023)] for the Brownian plane. An important notion of Section 4 is a slice which separates two successive disjoint geodesics from the hull boundary to the ball of radius $$\varepsilon$$. Section 5 uses the results of Section 3 to derive the key estimate in Lemma 15. Section 6 then gives the proof of Theorem 1 along the lines of the preceding discussion. The Appendix contains the proofs of a couple of technical lemmas, including the strong coupling between the Brownian plane and the Brownian sphere that is used to justify the zero-one law argument.

### MSC:

 60D05 Geometric probability and stochastic geometry 53C22 Geodesics in global differential geometry 53C65 Integral geometry

### Citations:

Zbl 1278.60124; Zbl 1132.60013; Zbl 1407.60018; Zbl 1480.60023
Full Text:

### References:

 [1] ABRAHAM, C. (2016). Rescaled bipartite planar maps converge to the Brownian map. Ann. Inst. Henri Poincaré Probab. Stat. 52 575-595. · Zbl 1375.60034 · doi:10.1214/14-AIHP657 [2] Abraham, C. and Le Gall, J.-F. (2018). Excursion theory for Brownian motion indexed by the Brownian tree. J. Eur. Math. Soc. (JEMS) 20 2951-3016. · Zbl 1501.60046 · doi:10.4171/JEMS/827 [3] ABRAHAM, R., DELMAS, J.-F. and HOSCHEIT, P. (2013). A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18 no. 14, 21. · Zbl 1285.60004 · doi:10.1214/EJP.v18-2116 [4] ADDARIO-BERRY, L. and ALBENQUE, M. (2017). The scaling limit of random simple triangulations and random simple quadrangulations. Ann. Probab. 45 2767-2825. · Zbl 1417.60022 · doi:10.1214/16-AOP1124 [5] ANGEL, O., KOLESNIK, B. and MIERMONT, G. (2017). Stability of geodesics in the Brownian map. Ann. Probab. 45 3451-3479. · Zbl 1407.60018 · doi:10.1214/16-AOP1140 [6] BETTINELLI, J., JACOB, E. and MIERMONT, G. (2014). The scaling limit of uniform random plane maps, via the Ambjørn-Budd bijection. Electron. J. Probab. 19 no. 74, 16. · Zbl 1320.60088 · doi:10.1214/EJP.v19-3213 [7] Bettinelli, J. and Miermont, G. (2017). Compact Brownian surfaces I: Brownian disks. Probab. Theory Related Fields 167 555-614. · Zbl 1373.60062 · doi:10.1007/s00440-016-0752-y [8] Curien, N. and Le Gall, J.-F. (2014). The Brownian plane. J. Theoret. Probab. 27 1249-1291. · Zbl 1305.05208 · doi:10.1007/s10959-013-0485-0 [9] CURIEN, N. and LE GALL, J.-F. (2016). The hull process of the Brownian plane. Probab. Theory Related Fields 166 187-231. · Zbl 1347.05226 · doi:10.1007/s00440-015-0652-6 [10] GWYNNE, E. Geodesic networks in Liouville quantum gravity surfaces. Preprint. Available at arXiv:2010.11260. · Zbl 1489.60016 [11] GWYNNE, E. and MILLER, J. (2017). Scaling limit of the uniform infinite half-plane quadrangulation in the Gromov-Hausdorff-Prokhorov-uniform topology. Electron. J. Probab. 22 Paper No. 84, 47. · Zbl 1378.60030 · doi:10.1214/17-EJP102 [12] Gwynne, E. and Miller, J. (2020). Confluence of geodesics in Liouville quantum gravity for $\gamma \in (0,2)$. Ann. Probab. 48 1861-1901. · Zbl 1453.60141 · doi:10.1214/19-AOP1409 [13] Gwynne, E. and Miller, J. (2021). Existence and uniqueness of the Liouville quantum gravity metric for $\gamma \in (0,2)$. Invent. Math. 223 213-333. · Zbl 1461.83018 · doi:10.1007/s00222-020-00991-6 [14] GWYNNE, E., PFEFFER, J. and SHEFFIELD, S. Geodesics and metric ball boundaries in Liouville quantum gravity. Preprint. Available at arXiv:2010.07889. [15] Le Gall, J.-F. (1999). Spatial Branching Processes, Random Snakes and Partial Differential Equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel. · Zbl 0938.60003 · doi:10.1007/978-3-0348-8683-3 [16] LE GALL, J.-F. (2007). The topological structure of scaling limits of large planar maps. Invent. Math. 169 621-670. · Zbl 1132.60013 · doi:10.1007/s00222-007-0059-9 [17] LE GALL, J.-F. (2010). Geodesics in large planar maps and in the Brownian map. Acta Math. 205 287-360. · Zbl 1214.53036 · doi:10.1007/s11511-010-0056-5 [18] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880-2960. · Zbl 1282.60014 · doi:10.1214/12-AOP792 [19] LE GALL, J.-F. (2015). Bessel processes, the Brownian snake and super-Brownian motion. In In Memoriam Marc Yor—Séminaire de Probabilités XLVII. Lecture Notes in Math. 2137 89-105. Springer, Cham. · Zbl 1334.60174 · doi:10.1007/978-3-319-18585-9_5 [20] LE GALL, J.-F. (2018). Subordination of trees and the Brownian map. Probab. Theory Related Fields 171 819-864. · Zbl 1405.60128 · doi:10.1007/s00440-017-0794-9 [21] Le Gall, J.-F. (2019). Brownian disks and the Brownian snake. Ann. Inst. Henri Poincaré Probab. Stat. 55 237-313. · Zbl 1466.60021 · doi:10.1214/18-aihp882 [22] LE GALL, J.-F. and RIERA, A. (2021). Spine representations for non-compact models of random geometry. Probab. Theory Related Fields 181 571-645. · Zbl 1480.60023 · doi:10.1007/s00440-021-01069-x [23] LE GALL, J.-F. and WEILL, M. (2006). Conditioned Brownian trees. Ann. Inst. Henri Poincaré Probab. Stat. 42 455-489. · Zbl 1107.60053 · doi:10.1016/j.anihpb.2005.08.001 [24] MARZOUK, C. (2018). Scaling limits of random bipartite planar maps with a prescribed degree sequence. Random Structures Algorithms 53 448-503. · Zbl 1397.05164 · doi:10.1002/rsa.20773 [25] MIERMONT, G. (2009). Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42 725-781. · Zbl 1228.05118 · doi:10.24033/asens.2108 [26] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319-401. · Zbl 1278.60124 · doi:10.1007/s11511-013-0096-8 [27] MILLER, J. and QIAN, W. Personal communication. [28] MILLER, J. and QIAN, W. (2020). Geodesics in the Brownian map: Strong confluence and geometric structure. Preprint. Available at arXiv:2008.02242. [29] Serlet, L. (1997). A large deviation principle for the Brownian snake. Stochastic Process. Appl. 67 101-115 · Zbl 0889.60026 · doi:10.1016/S0304-4149(97)00128-7
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