On an open interval we follow the paths of a Brownian motion which returns to a fixed point as soon as it reaches the boundary and restarts afresh indefinitely. We determine that two paths starting at different points either cannot collapse or they do so almost surely. The problem can be modelled as a spatially inhomogeneous random walk on a group and contrasts sharply with the higher dimension...

This paper describes a graph-spectral method for path estimation. Our aim is to find a maximum probability path through a lattice of pixel sites. We characterise the path recovery problem using a site transition matrix. A graph-spectral analysis of the transition matrix reveals how the maximum probability path can be located using an eigenvector of the associated normalised affinity matrix. We ...

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the β-Hermite and β-Laguerre ensembles of random matrix theory. We conclude by p...

We consider the problem of generating permutations almost uniformly at random in distributed and parallel systems. We propose a simple distributed scheme for permuting at random, which we call distributed mixing, and provide its precise stochastic analysis. Our main result is that distributed mixing needs log n simple point-to-point communication rounds to generate a permutation almost uniforml...

We study distance properties of a general class of random directed acyclic graphs (DAGs). In a DAG, many natural notions of distance are possible, for there exists multiple paths between pairs of nodes. The distance of interest for circuits is the maximum length of a path between two nodes. We give laws of large numbers for the typical depth (distance to the root) and the minimum depth in a ran...

Several randomized path planners have been proposed during the last few years. Their attractiveness stems from their applicability to virtually any type of robots, and their empirically observed success. In this paper we attempt to present a unifying view of these planners and to theoretically explain their success. First, we introduce a general planning scheme that consists of randomly samplin...

The purpose of this article is to bring together some results on random assignment and shortest path problems, and to clarify how they relate to each other. In particular, we investigate the consequences of the two very different and hard proofs of the ζ(2) limit theorem for the random assignment problem given by David Aldous [1] and by Chandra Nair, Balaji Prabhakar and Mayank Sharma [11]. The...

The shortest path problem is one of the most fundamental problems in network optimization. This paper is concerned with shortest path problems in non-deterministic environment, in which the lengths of some arcs have stochastic characteristics and meanwhile some have uncertain characteristics. In order to investigate paths in these networks, this paper introduces the chance distribution of minim...

Total path length, or search cost, for a rooted tree is defined as the sum of all root-to-node distances. Let Tn be the total path length for a random recursive tree of order n. Mahmoud (1991) showed that Wn := (Tn − E[Tn])/n converges almost surely and in L2 to a nondegenerate limiting random variable W . Here we give recurrence relations for the moments of Wn and of W and show that Wn converg...