arXiv research

Study random walks on groups with superlinear divergent geodesics.

Developed a random walk analog of geodesic flow on hyperbolic groups.

Geodesic walks converge to Brownian motion on Finsler manifolds.

Study geodesics on graphs with random lengths, proving bi-infinite paths exist.

Fold maps associated to geodesic random walks on curved spaces.

Study on geodesics and eigenvalues on random hyperbolic surfaces with cusps.

Study random walks on sub-Riemannian manifolds using retractions.

Random simple closed curves map Teichmüller space to geodesic currents.

Study on random surfaces in hyperbolic 3-manifolds, focusing on geometric and topological properties.

The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.

New method calculates geodesic distances in Gaussian random field manifolds.

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a …

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

Random walks on mapping class groups identified with geodesic laminations.

We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into …

Study on geodesics on random hyperbolic surfaces, showing variance asymptotic to X log X.

We investigate the distribution of lengths obtained by intersecting a random geodesic with a geodesic lamination. We give an explicit formula for the distribution for the case of a maximal lamination and show that the distribution is independent of the surface and lamination. We also show how the moments of the distrib…

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolut…

Study on random hyperbolic surfaces with many cusps, focusing on tight geodesics.

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmuller geodesic is in the principal stratum of quadratic differentials. We show that a Teichmuller ge…

Formula for integrating random variables on hyperbolic surfaces.

The minimal area of Finsler disks with minimizing geodesics is at least 6/π r^2.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

We consider random walks on the mapping class group whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anoso…

This is supplementary material for the main Geodesics article by the authors. In Appendix A, we present some general results on the construction of Gaussian random fields. In Appendix B, we restate our Shape Theorem, specialized to the setting of this article. In Appendix C, we state some straightforward consequences o…

Study geodesics on random hyperbolic surfaces, finding variance similar to prime number theory.

We show that, in the Teichmüller metric, "thin-framed triangles are thin"---that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on Teichmüller space. In particular, an application is worked out for the action of th…

We show that simple random walks on (non-trivial) relatively hyperbolic groups stay $O(\log(n))$-close to geodesics, where $n$ is the number of steps of the walk. Using similar techniques we show that simple random walks in mapping class groups stay $O(\sqrt{n\log(n)})$-close to geodesics and hierarchy paths. Along the…

Study geodesic trees and exceptional directions in FPP on hyperbolic groups.

For a non-uniform lattice in SL(2,R), we consider excursions in cusp neighborhoods of a random geodesic on the corresponding finite area hyperbolic surface or orbifold. We prove a strong law for a certain partial sum involving these excursions. This generalizes a theorem of Diamond and Vaaler for continued fractions. I…

Topology helps estimate chromatic numbers of random graphs on spheres.

Random 3-manifolds have no totally geodesic submanifolds.

The study proves nearly geodesic surfaces are filling in hyperbolic 3-manifolds.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

In this work we study the degree distribution, the maximum vertex and edge flow in non-uniform random Delaunay triangulations when geodesic routing is used. We also investigate the vertex and edge flow in Erdös-Renyi random graphs, geometric random graphs, expanders and random $k$-regular graphs. Moreover we show that …

FPP preserves sublinear Morse boundaries in geodesic graphs.

We prove a central limit theorem for the length of closed geodesics in any compact orientable hyperbolic surface. In the special case of a hyperbolic pair of pants, this settles a conjecture of Chas-Li-Maskit.

New Teichmüller geodesic rays found with unique foliations.

Study of large-$n$ asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.

We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the …

The paper calculates how random changes affect paths on a complex geometric space.

Study shortest geodesics on flat cone spheres with conical singularities.