Stability of geodesics in the Brownian map

Abstract

The Brownian map is a random geodesic metric space arising as the scaling limit of random planar maps. We strengthen the so-called confluence of geodesics phenomenon observed at the root of the map, and with this, reveal several properties of its rich geodesic structure.Our main result is the continuity of the cut locus at typical points. A small shift from such a point results in a small, local modification to the cut locus. Moreover, the cut locus is uniformly stable, in the sense that any two cut loci coincide outside a closed, nowhere dense set of zero measure.We obtain similar stability results for the set of points inside geodesics to a fixed point. Furthermore, we show that the set of points inside geodesics of the map is of first Baire category. Hence, most points in the Brownian map are endpoints.Finally, we classify the types of geodesic networks which are dense. For each k ∈ {1, 2, 3, 4, 6, 9}, there is a dense set of pairs of points which are joined by networks of exactly k geodesics and of a specific topological form. We find the Hausdorff dimension of the set of pairs joined by each type of network. All other geodesic networks are nowhere dense.

Existence and uniqueness of the Liouville quantum gravity metric for $$\gamma \in (0,2)$$

“…Is this number finite, and, if so, does it depend on γ ? More generally, can one prove results about the possible topologies of the set of γ -LQG geodesics joining two points in C analogous to the results for the Brownian map in [3]?…”

“…If h is a whole-plane GFF plus a bounded continuous function, we define h * ε (z) for ε > 0 and z ∈ C as in (1.2) for our given choice of h. For z, w ∈ C and ε > 0, we define the ε-LFPP metric by D ε h (z, w) := inf P:z→w 1 0 e ξ h * ε (P(t)) |P (t)| dt (1.6) where the infimum is over all piecewise continuously differentiable paths from z to w. One should think of LFPP as the metric analog of the approximations of the LQG measure in (1.3). 3 The intuitive reason why we look at e ξ h * ε (z)…”

“…By (1.6), this results in scaling distances by C ξ/γ = C 1/d γ , which is consistent with the 2 The reason why we sometimes restrict to bounded continuous functions is to ensure that the convolution with the whole-plane heat kernel is finite (so D ε h is defined) and that the results about subsequential limits of LFPP in [16,18] are applicable. 3 One can also consider other variants of LFPP, defined using different approximations of the GFF, but we consider h * ε here since this is the approximation for which tightness is proven in [16]. If we knew tightness and some basic properties of the subsequential limiting metrics for LFPP defined using a different approximation of the GFF, then Theorem 1.8 below would show that these variants of LFPP also converge to the γ -LQG metric.…”

“…We do not show that P actually merges into the D h -geodesic from u to v. We believe that it should be possible to show that P merges into this D h -geodesic, but doing so is highly non-trivial. Indeed, this is closely related to the problem of showing that there are no "ghost geodesics" for D h which do not merge into any other D h -geodesics; see [3,Section 1.4] for some discussion about the analogous problem in the setting of the Brownian map. Because we do not show that P merges into the D h -geodesic from u to v, the arguments of this section do not immediately imply other statements of the form "if an event occurs for some (random) geodesic with high probability, then with high probability it occurs somewhere along the D h -geodesic between two fixed points".…”

Infinite geodesics in hyperbolic random triangulations

“…[5] Almost surely, for any x ∈ m ∞ , if γ and γ ′ are two geodesics from ρ to x that coincide on a neighbourhood of x, then γ = γ ′ .…”