978 research outputs found
Convergence in a multidimensional randomized Keynesian beauty contest
We study the asymptotics of a Markovian system of particles in
in which, at each step in discrete time, the particle farthest from
the current centre of mass is removed and replaced by an independent random particle. We show that the limiting configuration contains
coincident particles at a random location . A key tool
in the analysis is a Lyapunov function based on the squared radius of gyration
(sum of squared distances) of the points. For d=1 we give additional results on
the distribution of the limit , showing, among other things, that it
gives positive probability to any nonempty interval subset of , and
giving a reasonably explicit description in the smallest nontrivial case, N=3.Comment: 26 pages, 4 figure
Limiting behaviour of random spatial graphs and asymptotically homogeneous RWRE
We consider several random spatial graphs of the nearest-neighbour type, including the k- nearest neighbours graph, the on-line nearest-neighbour graph, and the minimal directed spanning tree. We study the large sample asymptotic behaviour of the total length of these graphs, with power-weighted edges. We give laws of large numbers and weak convergence results. We evaluate limiting constants explicitly. In Bhatt and Roy's minimal directed spanning tree (MDST) construction on random points in (0,1)(^2), each point is joined to its nearest neighbour in the south-westerly direction. We show that the limiting total length (with power-weighted egdes) of the edges joined to the origin converges in distribution to a Dickman-type random variable. We also study the length of the longest edge in the MDST. For the total weight of the MDST, we give a weak convergence result. The limiting distribution is given a normal component plus a contribution due to boundary effects, which can be characterized by a fixed point equation. There is a phase transition in the limit law as the weight exponent increases. In the second part of this thesis, we give criteria for ergodicity, transience and null recurrence for the random walk in random environment (RWRE) on z+ = {0,1,2,...}, with reflection at the origin, where the random environment is subject to a vanishing perturbation from the so-called Sinai's regime. Our results complement existing criteria for random walks in random environments and for Markov chains with asymptotically zero drift, and are significantly different to these previously studied cases. Our method is based on a martingale technique 一 the method of Lyapunov functions
Deposition, diffusion, and nucleation on an interval
Motivated by nanoscale growth of ultra-thin films, we study a model of deposition, on an interval substrate, of particles that perform Brownian motions until any two meet, when they nucleate to form a static island, which acts as an absorbing barrier to subsequent particles. This is a continuum version of a lattice model studied in the applied literature. We show that the associated interval-splitting process converges in the sparse deposition limit to a Markovian process (in the vein of Brennan and Durrett) governed by a splitting density with a compact Fourier series expansion but, apparently, no simple closed form. We show that the same splitting density governs the fixed deposition rate, large time asymptotics of the normalized gap distribution, so these asymptotics are independent of deposition rate. The splitting density is derived by solving an exit problem for planar Brownian motion from a right-angled triangle, extending work of Smith and Watson
Logarithmic speeds for one-dimensional perturbed random walks in random environments
We study the random walk in a random environment on Z+={0,1,2,…}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai’s regime; (ii) a simple random walk with a random perturbation. We give almost sure results on how far the random walker is from the origin, for almost every environment. We give both upper and lower almost sure bounds. These bounds are of order (logt)β, for β∈(1,∞), depending on the perturbation. In addition, in the ergodic cases, we give results on the rate of decay of the stationary distribution
Non-homogeneous random walks on a semi-infinite strip
We study the asymptotic behaviour of Markov chains (Xn,ηn) on Z+×S, where Z+ is the non-negative integers and S is a finite set. Neither coordinate is assumed to be Markov. We assume a moments bound on the jumps of Xn, and that, roughly speaking, ηn is close to being Markov when Xn is large. This departure from much of the literature, which assumes that ηn is itself a Markov chain, enables us to probe precisely the recurrence phase transitions by assuming asymptotically zero drift for Xn given ηn. We give a recurrence classification in terms of increment moment parameters for Xn and the stationary distribution for the large- X limit of ηn. In the null case we also provide a weak convergence result, which demonstrates a form of asymptotic independence between Xn (rescaled) and ηn. Our results can be seen as generalizations of Lamperti’s results for non-homogeneous random walks on Z+ (the case where S is a singleton). Motivation arises from modulated queues or processes with hidden variables where ηn tracks an internal state of the system
Limit theorems for random spatial drainage networks
Suppose that, under the action of gravity, liquid drains through the unit d-cube via a minimal-length network of channels constrained to pass through random sites and to flow with nonnegative component in one of the canonical orthogonal basis directions of Rd, d ≥ 2. The resulting network is a version of the so-called minimal directed spanning tree. We give laws of large numbers and convergence in distribution results on the large-sample asymptotic behaviour of the total power-weighted edge length of the network on uniform random points in (0, 1)d. The distributional results exhibit a weight-dependent phase transition between Gaussian and boundary-effect-derived distributions. These boundary contributions are characterized in terms of limits of the so-called on-line nearest-neighbour graph, a natural model of spatial network evolution, for which we also present some new results. Also, we give a convergence in distribution result for the length of the longest edge in the drainage network; when d = 2, the limit is expressed in terms of Dickman-type variables
On the total length of the random minimal directed spanning tree
In Bhatt and Roy's minimal directed spanning tree construction for a random, partially ordered set of points in the unit square, all edges must respect the `coordinatewise' partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power-weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary plus a contribution introduced by the boundary effects, which can be characterized by a fixed-point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects being dominant. In the critical case in which the weight is simple Euclidean length, both effects contribute significantly to the limit law
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