Geodesics in the Brownian map : strong confluence and geometric structure / by Jason Miller, and Wei Qian.

Format:

Book

Author/Creator:

Miller, Jason, 1983- author.

Contributor:

Qian, Wei (Mathematician)

Series:

Memoirs of the American Mathematical Society ; vol.315, no. 1602.

Memoirs of the American Mathematical Society, 0065-9266 ; vol. 315, no. 1602

Language:

English

Subjects (All):

Geodesics (Mathematics).

Metric spaces.

Measure theory.

Physical Description:

vi, 117 pages : illustrations, 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2025.

Summary:

We study geodesics in the Brownian map (S, d, v), the random metric measure space which arises as the Gromov-Hausdorff scaling limit of uniformly random planar maps. Our results apply to all geodesics including those between exceptional points. First, we prove a strong and quantitative form of the confluence of geodesics phenomenon which states that any pair of geodesics which are sufficiently close in the Hausdorff distance must coincide with each other except near their endpoints. Then, we show that the intersection of any two geodesics minus their endpoints is connected, the number of geodesics which emanate from a single point and are disjoint except at their starting point is at most 5, and the maximal number of geodesics which connect any pair of points is 9. For each 1≤k≤9, we obtain the Hausdorff dimension of the pairs of points connected by exactly k geodesics. For k=7, 8, 9, such pairs have dimension zero and are countably infinite. Further, we classify the (finite number of) possible configurations of geodesics between any pair of points in S, up to homeomorphism, and give a dimension upper bound for the set of endpoints in each case. Finally, we show that every geodesic can be approximated arbitrarily well and in a strong sense by a geodesic connecting v-typical points. In particular, this gives an affirmative answer to a conjecture of Angel, Kolesnik, and Miermont that the geodesic frame of S, the union of all of the geodesics in S minus their endpoints, has dimension one, the dimension of a single geodesic.

Contents:

1. Introduction

2. Preliminaries

3. Close geodesics must intersect near their endpoints

4. Finite number of geodesics

5. Proof of strong confluence and topology of geodesics

6. Exponent for disjoint geodesics from a point

7. Completion of proofs of dimension upper bounds

8. Points connected by 5, 7, and 8 geodesics.

Notes:

"Number 1602 (sixth of 6 numbers)"--Cover.

Includes bibliographical references.

ISBN:

1470476150

9781470476151

OCLC:

1560084973