Related papers: Sublinear deviation between geodesics and sample p…

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key…

We continue the comparison between lines of minima and Teichmueller geodesics begun in [CRS1]. We show that in the Teichmueller space of a surface S, lines of minima are quasi-geodesic with respect to the Teichmueller metric. The…

We use a one-dimensional random walk on $D$-dimensional hyper-spheres to determine the critical behavior of statistical systems in hyper-spherical geometries. First, we demonstrate the properties of such walk by studying the phase diagram…

We prove effective density of random walks on homogeneous spaces, assuming that the underlying measure is supported on matrices generating a dense subgroup and having algebraic entries. The main novelty is an argument passing from high…

In this article, we develop a theory for understanding the traces left by a random walk in the vicinity of a randomly chosen reference vertex. The analysis is related to interlacements but goes beyond previous research by showing weak limit…

We consider a random walk on a homogeneous space $G/\Lambda$ where $G$ is $\mathrm{SO}(2,1)$ or $\mathrm{SO}(3,1)$ and $\Lambda$ is a lattice. The walk is driven by a probability measure $\mu$ on $G$ whose support generates a Zariski-dense…

In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain…

Random walks represent an important tool for probing the structural and dynamical properties of networks and modeling transport and diffusion processes on networks. However, when individuals' movement becomes dictated by more complicated…

In this work, we consider an inhomogeneous (discrete time) Markov chain and are interested in its long time behavior. We provide sufficient conditions to ensure that some of its asymptotic properties can be related to the ones of a…

In this paper we establish a version of the Margulis Roblin equidistribution theorem's for harmonic measures. As a consequence a von Neumann type theorem is obtained for boundary actions and the irreducibility of the associated…

We study a one-parameter family of discrete-time quantum walk models on the line and in the xy-plane associated with the Hadamard walk. Weak convergence in the long-time limit of all moments of the walker's pseudo-velocity on the line and…

We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line. We also study the continuous limit of this model, and discuss in detail the relation between…

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on…

This investigation is motivated by a result we proved recently for the random transposition random walk: the distance from the starting point of the walk has a phase transition from a linear regime to a sublinear regime at time $n/2$. Here,…

We consider compact Grassmann manifolds $G/K$ over the real, complex or quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of type $BC$. From an explicit integral representation of these polynomials we deduce a…

Li-Nadler proposed a conjecture about traces of Hecke categories, which implies the semistable part of the Betti Geometric Langlands Conjecture of Ben-Zvi-Nadler in genus 1. We prove a Weyl group analogue of this conjecture. Our theorem…

We provide an easy approach to the geodesic distance on the general linear group GL(n) for left-invariant Riemannian metrics which are also right-O(n)-invariant. The parametrization of geodesic curves and the global existence of length…

We show that the subsurface projection of a train track splitting sequence is an unparameterized quasi-geodesic in the curve complex of the subsurface. For the proof we introduce induced tracks, efficient position, and wide curves. This…

In this article we prove some previously announced results about metric ultraproducts of finite simple groups. We show that any non-discrete metric ultraproduct of alternating or special linear groups is a geodesic metric space. For more…

The aim of this work is to study the convergence to equilibrium of an $(h,\rho)$-subelliptic random walk on a closed, connected Riemannian manifold $(M,g)$ associated with a subelliptic second-order differential operator $A$ on $M$. In such…