Andrew R. Wade

RANDOM MINIMAL DIRECTED SPANNING TREES AND DICKMAN-TYPE DISTRIBUTIONS

In Bhatt and Roys minimal directed spanning tree construction for n random points in the unit square, all edges must be in a south-westerly direction and there must be a directed path from each vertex to the root placed at the origin. We identify the limiting distributions (for large n) for the total length of rooted edges, and also for the maximal length of all edges in the tree. These limit distributions have been seen previously in analysis of the Poisson-Dirichlet distribution and elsewhere; they are expressed in terms of Dickmans function, and their properties are discussed in some detail.

On the total length of the random minimal directed spanning tree

In Bhatt and Roys 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.

Explicit laws of large numbers for random nearest-neighbour-type graphs.

Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability 13 (2003), 277-303) we give laws of large numbers (in the L p sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝ d , d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

Asymptotic behaviour of randomly reflecting billiards in unbounded tubular domains

We study stochastic billiards in infinite planar domains with curvilinear boundaries: that is, piecewise deterministic motion with randomness introduced via random reflections at the domain boundary. Physical motivation for the process originates with ideal gas models in the Knudsen regime, with particles reflecting off microscopically rough surfaces. We classify the process into recurrent and transient cases. We also give almost-sure results on the long-term behaviour of the location of the particle, including a super-diffusive rate of escape in the transient case. A key step in obtaining our results is to relate our process to an instance of a one-dimensional stochastic process with asymptotically zero drift, for which we prove some new almost-sure bounds of independent interest. We obtain some of these bounds via an application of general semimartingale criteria, also of some independent interest.

Multivariate normal approximation in geometric probability

Consider a measure μλ = Σxξxδx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in ℝd are asymptotically independent normals as λ → ℞ here we give an O(λ−1/(2d+ε)) bound on the rate of convergence, and also a new criterion for the limiting normals to be non-degenerate. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

Rank-driven Markov processes

We study a class of Markovian systems of N elements taking values in [0,1] that evolve in discrete time t via randomized replacement rules based on the ranks of the elements. These rank-driven processes are inspired by variants of the Bak–Sneppen model of evolution, in which the system represents an evolutionary ‘fitness landscape’ and which is famous as a simple model displaying self-organized criticality. Our main results are concerned with long-time large-N asymptotics for the general model in which, at each time step, K randomly chosen elements are discarded and replaced by independent U[0,1] variables, where the ranks of the elements to be replaced are chosen, independently at each time step, according to a distribution κN on {1,2,…,N}K. Our main results are that, under appropriate conditions on κN, the system exhibits threshold behavior at s∗∈[0,1], where s∗ is a function of κN, and the marginal distribution of a randomly selected element converges to U[s∗,1] as t→∞ and N→∞. Of this class of models, results in the literature have previously been given for special cases only, namely the ‘mean-field’ or ‘random neighbor’ Bak–Sneppen model. Our proofs avoid the heuristic arguments of some of the previous work and use Foster–Lyapunov ideas. Our results extend existing results and establish their natural, more general context. We derive some more specialized results for the particular case where K=2. One of our technical tools is a result on convergence of stationary distributions for families of uniformly ergodic Markov chains on increasing state-spaces, which may be of independent interest.

Asymptotic theory for the multidimensional random on-line nearest-neighbour graph

The on-line nearest-neighbour graph on a sequence of n uniform random points in (0,1)d () joins each point after the first to its nearest neighbour amongst its predecessors. For the total power-weighted edge-length of this graph, with weight exponent [alpha][set membership, variant](0,d/2], we prove O(max{n1-(2[alpha]/d),logn}) upper bounds on the variance. On the other hand, we give an n-->[infinity] large-sample convergence result for the total power-weighted edge-length when [alpha]>d/2. We prove corresponding results when the underlying point set is a Poisson process of intensity n.

Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

We study quantitative asymptotics of planar random walks that are spatially non-homogeneous but whose mean drifts have some regularity. Specifically, we study the first exit time τα from a wedge with apex at the origin and interior half-angle α by a non-homogeneous random walk on ℤ2 with mean drift at x of magnitude O(∥x∥−1) as ∥x∥→∞. This is the critical regime for the asymptotic behaviour: under mild conditions, a previous result of the authors stated that τα<∞ a.s. for any α. Here we study the more difficult problem of the existence and non-existence of moments

The simple harmonic urn.

{\mathbb{E}}[ \tau_{\alpha}^{s}]

Logarithmic speeds for one-dimensional perturbed random walks in random environments

, s>0. Assuming a uniform bound on the walk’s increments, we show that for αs0; under specific assumptions on the drift field, we show that we can attain