Related papers: Statistics of geodesics in large quadrangulations

Mixed effect modeling for longitudinal data is challenging when the observed data are random objects, which are complex data taking values in a general metric space without linear structure. In such settings the classical additive error…

In many singular metric spaces, the regularity of a shortest-length curve is unknown. Algebraic varieties, or more generally sets defined by finitely many polynomial or real analytic equalities or inequalities, all locally partition into…

We study the behaviour of geodesics on a Riemannian manifold near a generalized conical or cuspidal singularity. We show that geodesics entering a small neighbourhood of the singularity either hit the singularity or approach it to a…

Most phylogenetic analyses result in a sample of trees, but summarizing and visualizing these samples can be challenging. Consensus trees often provide limited information about a sample, and so methods such as consensus networks,…

For a compact surface $S$ with constant negative curvature $-\kappa$ (for some $\kappa>0$) and genus $g\geq2$, we show that the tails of the distribution of $i(\alpha,\beta)/l(\alpha)l(\beta)$ (where $i(\alpha,\beta)$ is the intersection…

In first-passage percolation, one places nonnegative i.i.d. random variables (T (e)) on the edges of Z d. A geodesic is an optimal path for the passage times T (e). Consider a local property of the time environment. We call it a pattern. We…

In this paper we show that on a complete Riemannian manifold of negative curvature and dimension $n>1$ every two points which realize a local maximum for the distance function are connected by at least $2n+1$ geometrically distinct geodesic…

In this paper we compare the different phenomena that occur when intersecting geometric objects with random geodesics on the unit sphere and inside convex bodies. On the high dimensional sphere we see that with probability bounded away from…

We derive the geodesic equation for relatively K\"ahler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log-norm…

In this article, we prove an extreme value theorem on the limit distribution of geodesics in a geometrically finite quotient of $\Gamma\backslash\mathcal{T}$ a locally finite tree. Main examples of such graphs are quotients of a Bruhat-Tits…

We investigate typical behavior of geodesics on a closed flat surface $S$ of genus $g\geq 2$. We compare the length quotient of long arcs in the same homotopy class with fixed endpoints for the flat and the hyperbolic metric in the same…

We give an accessible introduction and elaboration on the methods used in obtaining a geodesic, which is the curve of shortest length connecting two points lying on the surface of a function. This is found through computing what's known as…

For stationary first passage percolation in two dimensions, the existence and uniqueness of semi-infinite geodesics directed in particular directions or sectors has been considered by Damron and Hanson (Commun. Math. Phys., 2014), Ahlberg…

Geodesic distance, commonly called shortest path length, has proved useful in a great variety of disciplines. It has been playing a significant role in search engine at present and so attracted considerable attention at the last few…

We propose the concepts of vicinal mappings and firmly vicinal mappings in metric spaces. We obtain fixed point and convergence theorems for these mappings in complete geodesic spaces with curvature bounded above by one and apply our…

A geodesic is a shortest path which connects a pair of vertices of a graph G. In this paper we define the geodesic subpath number gpn(G) of a graph G as the number of geodesics in G. The number of subtrees and subpaths are already studied…

Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…

Considering the Teichm\"uller space of a surface equipped with Thurston's Lipschitz metric, we study geodesic segments whose endpoints have bounded combinatorics. We show that these geodesics are cobounded, and that the closest-point…

The two-dimensional surface of a bi-axial ellipsoid is characterized by the lengths of its major and minor axes. Longitude and latitude span an angular coordinate system across. We consider the egg-shaped surface of constant altitude above…

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual…