arXiv research

Study random walks on groups with superlinear divergent geodesics.

Geodesic walks converge to Brownian motion on Finsler manifolds.

Study random walks on sub-Riemannian manifolds using retractions.

Fold maps associated to geodesic random walks on curved spaces.

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

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

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

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 …

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…

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

Random walks on free groups reveal asymmetric expansion factors.

Deviation inequalities and limit laws for random walks on metric spaces.

Study large deviations and speed of random walks in hyperbolic spaces.

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…

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…

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…

Study examines large deviations in random walks on hyperbolic spaces.

We introduce the geodesic walk for sampling Riemannian manifolds and apply it to the problem of generating uniform random points from polytopes in R^n specified by m inequalities. The walk is a discrete-time simulation of a stochastic differential equation (SDE) on the Riemannian manifold equipped with the metric induc…

We relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular at…

This paper refines bounds on random walk speed in Teichmüller space.

We give a proof of the sublinear tracking property for sample paths of random walks on various groups acting on spaces with hyperbolic-like properties. As an application, we prove sublinear tracking in Teichmueller distance for random walks on mapping class groups, and on Cayley graphs of a large class of finitely gene…

We show that the largest subsurface projection distance between a marking and its image under the nth step of a random walk grows logarithmically in n, with probability approaching 1 as n tends to infinity. Our setup is general and also applies to (relatively) hyperbolic groups and to $\mathrm{Out}(F_n)$. We then use t…

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

We establish spectral theorems for random walks on mapping class groups of connected, closed, oriented, hyperbolic surfaces, and on $\text{Out}(F_N)$. In both cases, we relate the asymptotics of the stretching factor of the diffeomorphism/automorphism obtained at time $n$ of the random walk to the Lyapunov exponent of …

A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$ converges to the boundary $\partial X$ with probability one when the step distribution supports two independent loxodromics. In particular, the random walk makes positive linear progress. Progress is known to be linear with exponential decay when …

The hitting measure is singular and has dimension less than 1 for cocompact Fuchsian groups.

New Teichmüller geodesic rays found with unique foliations.

The Cannon-Thurston map's measures become singular with respect to sphere measures.

The Cannon-Thurston map's pushed measures on the circle are singular with respect to sphere measures.

We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting nondegenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length $n$ is of order $n/\l…

Every pseudo-Anosov mapping class $\varphi$ defines an associated veering triangulation $τ_\varphi$ of a punctured mapping torus. We show that generically, $τ_\varphi$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesic…

Study large deviations in random walks on Lie groups.

We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few…

This paper presents VEC-NBT, a variation on the unsupervised graph clustering technique VEC, which improves upon the performance of the original algorithm significantly for sparse graphs. VEC employs a novel application of the state-of-the-art word2vec model to embed a graph in Euclidean space via random walks on the n…

Quantum walks blend patterns into splines when averaged.

Local limit theorem for random walks on hyperbolic groups with parabolic subgroups.

Random walks on cell complexes link to Laplacians and Novikov-Shubin invariants.

Study diffusions and random walks on hyperbolic spaces, focusing on their Martin boundaries.

Random walks on metric spaces embed quasi-isometrically into the space.

This work estimates edge weights of edge-reinforced random walks using observed data.

New proof shows rapid mixing for random walks on nilmanifolds.

Random walks on hyperbolic spaces show linear growth in translation lengths.

The paper examines random walks on metric spaces and finds commensurable subgroups.

Survey on random walks on mapping class groups and their properties.

EntroPath learns manifold geometry from diffusion paths.

Random walks on Fuchsian Schottky groups have harmonic measures with lower dimension.

UniNet efficiently learns network representations from large graphs.