arXiv research

Algorithms find second and third shortest non-trivial closed walks on surfaces.

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 $\mathbb{R}^3$…

Random walk constructs Morse functions on surfaces.

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…

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…

The paper finds braid representatives minimizing simple walks for knots.

Random walk speed on Teichmüller space is a proper function.

We consider a random walk on the mapping class group of a surface of finite type. We assume that the random walk is determined by a probability measure whose support is finite and generates a non-elementary subgroup $H$. We further assume that $H$ is not consisting only of lifts with respect to any one covering. Then w…

Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. W…

Proposes a method to improve graph embedding by removing least frequent nodes.

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 …

Random walks on braid groups are transient, with specific closure properties for certain braids.

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

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

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

We obtain a description of Poisson--Furstenberg boundaries for (random walks on) fundamental groups of compact graph-manifolds. Together with previously known results due to V.A. Kaimanovich and others, this allows one to obtain descriptions of Poisson--Furstenberg boundaries for fundamental groups of all closed 3-mani…

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

DERWENT learns paths for distant transfer learning via deep random walk.

By appealing to renewal theory we determine the equations that the mean exit time of a continuous-time random walk with drift satisfies both when the present coincides with a jump instant or when it does not. Particular attention is paid to the corrections ensuing from the non-Markovian nature of the process. We show t…

A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.

We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation sugge…

We show that for any integers k and g, with g at least two, there are infinitely many closed hyperbolic 3-manifolds which are integral homology spheres with Casson invariant k, and Heegaard genus equal to g. This existence result is shown using random methods, using a model of random 3-manifolds arising from random wal…

Graphs on surfaces have limits for complete walks, impacting ergodicity.

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

In this paper, we propose a new fuzzy clustering algorithm based on the mode-seeking framework. Given a dataset in $\mathbb{R}^d$, we define regions of high density that we call cluster cores. We then consider a random walk on a neighborhood graph built on top of our data points which is designed to be attracted by hig…

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…

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

We propose and analyze two new MCMC sampling algorithms, the Vaidya walk and the John walk, for generating samples from the uniform distribution over a polytope. Both random walks are sampling algorithms derived from interior point methods. The former is based on volumetric-logarithmic barrier introduced by Vaidya wher…

Study large deviations in random walks on Lie groups.

The paper introduces walks with jumps for modeling neuron activity in hyperbolic space.

Many problems in finance are related to first passage times. Among all of them, we chose three on which we contributed personally. Our first example relates Kolmogorov-Smirnov like goodness-of-fit tests, modified in such a way that tail events and core events contribute equally to the test (in the standard Kolmogorov-S…

Quantum walks blend patterns into splines when averaged.

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

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

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…

Study random walks on sub-Riemannian manifolds using retractions.

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.

New proof shows rapid mixing for random walks on nilmanifolds.

Quantum walks model financial returns with flexibility and asymmetry.

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

Study random walks on groups with superlinear divergent geodesics.

New walk extraction strategies improve node embeddings in KGs.

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

Researchers use information geometry to analyze and improve DRWs for node classification.

Geodesic walks converge to Brownian motion on Finsler manifolds.

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

Random walks on free groups reveal asymmetric expansion factors.