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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
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