In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

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 LN = αN √ T and the mean area AN = βN T for all T . The prefactors αN and βN , 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 ecological context in estimating the home range of a herd of animals with population size N .PACS numbers: 87.23.Cc Ecologists often need to estimate the home range of an animal or a group of animals, in particular for habitatconservation planning [1]. Home range of a group of animals simply means the two dimensional space over which they typically move around in search of food. There exist various methods to estimate this home range, based on the monitoring of the positions of the animals over a certain period of time [2]. One method consists in drawing the minimum convex polygon enclosing all monitored positions, called the convex hull. While this may seem simple minded, it remains, under certain circumstances, the best way to proceed [3]. The monitored positions, for one animal, will appear as the vertices of a path whose statistical properties will depend on the type of motion the animal is performing. In particular, during phases of food searching known as foraging, the monitored positions can be described as the vertices of a random walk in the plane [4,5]. For animals whose daily motion consists mainly in foraging, quantities of interest about their home range, such as its perimeter and area, can be estimated through the average perimeter and area of the convex hull of the corresponding random walk ( Fig. 1(a)). If the recorded positions are numerous (which might result from a very fine and/or long monitoring), the number of steps of the random walker becomes large and to a good approximation the trajectory of a discrete-time planar random walk (with finite variance of the step sizes) can be replaced by a continuous-time planar Brownian motion of a certain duration T .The home range of a single animal can thus be characterized by the mean perimeter and area of the convex hull of a planar Brownian motion of duration T starting at origin O. Both 'open' (where the endpoint of the path is free) and 'closed' paths (that are constrained to return to the origin in time T ) are of interest. The latter corresponds, for instance, to an animal returning every night to its nest after spending the day foraging in the surroundings. For an 'open' path, the mean perimeter L 1 = √ 8πT and the mean area A 1 = πT /2 are known in the mathematics literature [6,7,8]. For a 'closed' path, only the mean perimeter is known [9]: L 1 = π 3 T /2. In any given habitat, an animal is however hardly alone but they live in herds with typically a large population size N . To study their global home range via the convex hull model, one needs to study the convex hull of N planar Brownian mo...

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We derive , the joint probability density of the maximum ) , ( m t M P M and the time at which this maximum is achieved, for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over m t M , the marginal density is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain 'agreement formulae' that are encountered more generally in probabilistic studies of Brownian motion processes. ) ( m t P ) ( m t P 1

We calculate analytically the probability density P(tm) of the time tm at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density P(M,tm) of the maximum M and tm. In the driftless case, we find that P(tm) has power-law tails: P(tm)∼tm−3/2 for large tm and P(tm)∼tm−1/2 for small tm. In the presence of a drift towards the origin, P(tm) decays exponentially for large tm. The results from numerical simulations are in excellent agreement with our analytical predictions.

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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.

We combine the processes of resetting and first passage, resulting in first-passage resetting, where the resetting of a random walk to a fixed position is triggered by the first-passage event of the walk itself. In an infinite domain, first-passage resetting of isotropic diffusion is non-stationary, and the number of resetting events grows with time according to t . We analytically calculate the resulting spatial probability distribution of the particle, and also obtain the distribution by geometric-path decomposition. In a finite interval, we define an optimization problem that is controlled by first-passage resetting; this scenario is motivated by reliability theory. The goal is to operate a system close to its maximum capacity without experiencing too many breakdowns. However, when a breakdown occurs the system is reset to its minimal operating point. We define and optimize an objective function that maximizes reward for being close to the maximum level of operation and imposes a penalty for each breakdown. We also investigate extensions of this basic model, firstly to include a delay after each reset, and also to two dimensions. Finally, we study the growth dynamics of a domain in which the domain boundary recedes by a specified amount whenever the diffusing particle reaches the boundary, after which a resetting event occurs. We determine the growth rate of the domain for a semi-infinite line and a finite interval and find a wide range of behaviors that depend on how much recession occurs when the particle hits the boundary.