arXiv research

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 …

The study of record statistics of correlated series is gaining momentum. In this work, we study the records statistics of the time series of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent $α$ lyi…

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

Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

Researchers prove hitting measure singularity for most Fuchsian and Kleinian groups.

This paper studies node embeddings of networks, revealing their geometric properties.

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…

Randomized control methods improve asset pricing and performance analysis.

Innovative extensions to option pricing models using asymmetric Brownian motion and random walk approaches.

A closed equilateral random walk in 3-space is a selection of unit length vectors giving the steps of the walk conditioned on the assumption that the sum of the vectors is zero. The sample space of such walks with $n$ edges is the $(2n-3)$-dimensional Riemannian manifold of equilateral closed polygons in $\mathbb{R}^3$…

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

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…

Several known results, by Rivin, Calegari-Maher and Sisto, show that an element $φ_n\in Out(F_r)$, obtained after $n$ steps of a simple random walk on $Out(F_r)$, is fully irreducible with probability tending to 1 as $n\to\infty$. In this paper we construct a natural "train-track directed" random walk $\mathcal W$ on $…

Quantum walks blend patterns into splines when averaged.

For finitely supported random walks on finitely generated groups $G$ we prove that the identity map on $G$ extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This yields new results for relatively hyperbolic groups. Our key e…

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

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…

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.

Study random walks on groups with superlinear divergent geodesics.

Study random walks on sub-Riemannian manifolds using retractions.

Let $S=Γ\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $Γ\le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $Γ$. When sampling by an $n$-step random walk in $π_1(S) \cong Γ$ with each step given by an element…

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

Random walks on free groups reveal asymmetric expansion factors.

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

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

UniNet efficiently learns network representations from large graphs.

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

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

Uniform drift estimates found for random walks on graph products.

A random Heegaard splitting is a 3-manifold obtained by using a random walk of length n on the mapping class group as the gluing map between two handlebodies. We show that the joint distribution of random walks of length n and their inverses is asymptotically independent, and converges to the product of the harmonic an…

Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.

Geodesic walks converge to Brownian motion on Finsler manifolds.

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

Hypergraphs are used in machine learning to model higher-order relationships in data. While spectral methods for graphs are well-established, spectral theory for hypergraphs remains an active area of research. In this paper, we use random walks to develop a spectral theory for hypergraphs with edge-dependent vertex wei…

We extend some properties of random walks on hyperbolic groups to random walks on convergence groups. In particular we prove that if a convergence group $G$ acts on a compact metrizable space $M$ with the convergence property then we can provide $G\cup M$ with a compact topology such that random walks on $G$ converge a…

For any pseudo-Anosov diffeomorphism on a closed orientable surface $S$ of genus greater than one, it is known by the work of Bers and Thurston that the topological entropy agrees with the translation distance on the Teichmüller space with respect to the Teichmüller metric. In this paper, we consider random walks on th…

Repelling random walks improve graph-based sampling efficiency.

Higher-order proximity preserved network embedding has attracted increasing attention. In particular, due to the superior scalability, random-walk-based network embedding has also been well developed, which could efficiently explore higher-order neighborhoods via multi-hop random walks. However, despite the success of …

Abstract: Nonlinear random walk with distributionally robust transition probabilities.

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