Segregation through the multiscalar lens.

We introduce a mathematical framework that allows one to carry out multiscalar and multigroup spatial exploratory analysis across urban regions. By producing coefficients that integrate information across all scales and that are normalized with respect to theoretical maximally segregated configurations, this framework provides a practical and powerful tool for the comparative empirical analysis of urban segregation. We illustrate our method with a study of ethnic mixing in the Los Angeles metropolitan area.

Convex hull of N planar Brownian motions: exact results and an application to ecology.

We compute exactly the mean perimeter and area of the convex hull of N independent planar Brownian paths each of duration T, both for open and closed paths. We show that the mean perimeter =alpha N sqrt[T] and the mean area =beta(N)T for all T. The prefactors alpha N and beta N, computed exactly for all N, increase very slowly (logarithmically) with increasing N. This slow growth is a consequence of extreme value statistics and has interesting implications in an ecological context in estimating the home range of a herd of animals with a population size N.

Facets on the convex hull of d-dimensional Brownian and Lévy motion.

For stationary, homogeneous Markov processes (viz., Lévy processes, including Brownian motion) in dimension d≥3, we establish an exact formula for the average number of (d-1)-dimensional facets that can be defined by d points on the process's path. This formula defines a universality class in that it is independent of the increments' distribution, and it admits a closed form when d=3, a case which is of particular interest for applications in biophysics, chemistry, and polymer science. We also show that the asymptotical average number of facets behaves as 〈F_{T}^{(d)}〉∼2[ln(T/Δt)]^{d-1}, where T is the total duration of the motion and Δt is the minimum time lapse separating points that define a facet.

Universality and time-scale invariance for the shape of planar Lévy processes.

For a broad class of planar Markov processes, viz. Lévy processes satisfying certain conditions (valid, e.g., in the case of Brownian motion and Lévy flights), we establish an exact, universal formula describing the shape of the convex hull of sample paths. We show indeed that the average number of edges joining paths' points separated by a time lapse Δτ ∈ [Δτ(1),Δτ(2)] is equal to 2 ln(Δτ(2)/Δτ(1)), regardless of the specific distribution of the process's increments and regardless of its total duration T. The formula also exhibits invariance when the time scale is multiplied by any constant. Apart from its theoretical importance, our result provides insights regarding the shape of two-dimensional objects (e.g., polymer chains) modeled by the sample paths of stochastic processes generally more complex than Brownian motion. In particular, for a total time (or parameter) duration T, the average number of edges on the convex hull ("cut off" to discard edges joining points separated by a time lapse shorter than some Δτ < T) will be given by 2 ln(T/Δτ). Thus it will only grow logarithmically, rather than at some higher pace.

Exact distribution of the maximal height of p vicious walkers.

Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that approximately square root 2p (excursions) whereas approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.