arXiv research

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…

Study random walks on sub-Riemannian manifolds using retractions.

Geodesic walks converge to Brownian motion on Finsler manifolds.

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

The study analyzes convergence of random-walk embeddings in graph theory.

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

Let G be a countable group which acts by isometries on a separable, but not necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly hyperbolic if G contains two independent hyperbolic isometries. We show that a random walk on such G converges to the Gromov boundary almost surely. We apply the co…

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…

Invariance principle proved for lifted geodesic walks on Riemannian submersions.

UniNet efficiently learns network representations from large graphs.

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

This paper considers a classical question of approximation of Brownian motion by a random walk in the setting of a sub-Riemannian manifold $M$. To construct such a random walk we first address several issues related to the degeneracy of such a manifold. In particular, we define a family of sub-Laplacian operators natur…

New random walk results on rank one symmetric spaces.

The Dirichlet random walk on manifolds has a positive escape rate if the cover is non-amenable.

A random walk on a countable group $G$ acting on a metric space $X$ gives a characteristic called the drift which depends only on the transition probability measure $μ$ of the random walk. The drift is the `translation distance' of the random walk. In this paper, we prove that the drift varies continuously with the tra…

Data-driven methods link graphon limits to random walks and spectral clustering.

This paper considers a sequence of discrete-time random walk markets with a safe and a single risky investment opportunity, and gives conditions for the existence of arbitrages or free lunches with vanishing risk, of the form of waiting to buy and selling the next period, with no shorting, and furthermore for weak conv…

Study on a pinning model with random walk increments, showing convergence to a critical disordered pinning measure.

Transformers learn random walks optimally with gradient descent.

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 contact graph of a CAT(0) cubical complex has unbounded structure and a Gaussian CLT for random walks.

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

TSAW improves MCMC integral estimation with faster convergence.

We describe random walk boundaries (in particular, the Poisson--Furstenberg, or PF-boundary) for a vast family of groups in terms of the hyperbolic boundary of a special free subgroup. We prove that almost all trajectories of the random walk (with respect to an arbitrary nondegenerate measure on the group) converge to …

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 $\mathbb{R}^3$…

Uniformly random permutations converge to regular representation on surface groups.

We consider the symmetric exclusion process on suitable random grids that approximate a compact Riemannian manifold. We prove that a class of random walks on these random grids converge to Brownian motion on the manifold. We then consider the empirical density field of the symmetric exclusion process and prove that it …

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.

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.

We study random walks on groups of isometries of non-proper delta-hyperbolic spaces under the assumption that at least one element in the group satisfies Bestvina-Fujiwara's WPD condition. We show that in this case typical elements are WPD, and the Poisson boundary coincides with the Gromov boundary. Moreover, we show …

New proof shows rapid mixing for random walks on nilmanifolds.

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

We study the problem of finding the maximum of a function defined on the nodes of a connected graph. The goal is to identify a node where the function obtains its maximum. We focus on local iterative algorithms, which traverse the nodes of the graph along a path, and the next iterate is chosen from the neighbors of the…

Study random walks on groups with superlinear divergent geodesics.

New tuning rules for Metropolis algorithms derived from Bayesian large-sample asymptotics.

Extends CRR model with q-binomial random walks for asset pricing.

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

Random walks on free groups reveal asymmetric expansion factors.

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

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

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

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