The Greenberg-Hastings model (GHM) is a family of multitype cellular automata that emulate excitable media, exhibiting the nucleation and spiral formation characteristic of such complex systems. In this paper we study the asymptotic frequency of nucleation in GHM dynamics on $\mathbb{Z}^2$ as the number of types, or colors, becomes large. Starting from uniform product measure over $\kappa$ colors, and assuming that the excitation threshold $\theta$ is not too large, the box size $L_\kappa$ needed for formation of a spiral core is shown to grow exponentially: $L_\kappa \approx \exp\{C_\kappa\}$ as $\kappa \rightarrow \infty$. By exploiting connections with percolation theory, we find that $C = 0.23 \pm 0.06$ in the nearest neighbor, $\theta = 1$ case. In contrast, GHM rules obey power law nucleation scaling when started from a suitable nonuniform product measure over the $\kappa$ colors. This effect is driven by critical percolation. Finally, we present some analogous results for a random GHM, an interacting Markovian system closely related to the epidemic with regrowth of Durrett and Neuhauser.

Consider a job-shop or batch-flow manufacturing system in which new jobs are introduced only as old ones depart, either because of physical constraints or as a matter of management policy. Assuming that there is never a shortage of new work to be done, the number of active jobs remains constant over time, and the system can be modeled as a kind of closed queueing network. With manufacturing applications in mind, we formulate a general closed network model and develop a mathematical method to estimate its steady-state performance characteristics. A restrictive feature of our network model is that all the job classes that are served at any given node or station share a common service time distribution. Our analytical method, which is based on an algorithm for computing the stationary distribution of an approximating Brownian model, is motivated by heavy traffic theory; it is precisely analogous to a method developed earlier for analysis of open queueing networks. The required inputs include not only first-moment information, such as average product mix and average processing rates, but also second-moment data that serve as quantitative measures of variability in the processing environment. We present numerical examples that show that system performance is very much affected by changes in second-moment data. In these few numerical examples, our estimates of average throughput rates and average throughput times for different product families are generally accurate when compared against simulation results.

Nash obtained an important extension to the classical theory of Gittins indexation when he demonstrated that index policies were optimal for a class of multiarmed bandit problems with a multiplicatively separable reward structure. We characterise the relevant indices (herein referred to as Nash indices) as equivalent retirement rewards/penalties for appropriately defined maximisation/minimisation problems. We also give a condition which is sufficient to guarantee the optimality of index policies for a Nash-type model in which each constituent bandit has its own decision structure.

We prove central limit theorems and establish rates of convergence for the following problems in geometrical probability when points are generated in the $\lbrack 0,1\rbrack^2$ cube according to a Poisson point process with parameter $n$: 1. The length of the nearest graph $N_{k,n}$, in which each point is connected to its $k$th nearest neighbor. 2. The length of the Delaunay triangulation $\operatorname{Del}_n$ of the points. 3. The length of the Voronoi diagram $\operatorname{Vor}_n$ of the points. Using the technique of dependency graphs of Baldi and Rinott, we show that the dependence range in all these problems converges quickly to 0 with high probability. Our approach also establishes rates of convergence for the number of points in the convex hull and the area outside the convex hull for points generated according to a Poisson point process in a circle.

In this paper we study a model of the competition of annual and perennial plants proposed by Crawley and May. Specifically, we calculate the asymptotic behavior of the critical value for coexistence in the biologically reasonable limit in which annual seeds are dispersed over a large number of sites. Our results are related to earlier work of Durrett and Swindle and prove a conjecture made in that paper.

Let $Q^\alpha_n$ be the probability measure for an $n$-step random walk $(0,S_1,\ldots,S_n)$ on $\mathbb{Z}$ obtained by weighting simple random walk with a factor $1 - \alpha$ for every self-intersection. This is a model for a one-dimensional polymer. We prove that for every $\alpha \in (0,1)$ there exists $\theta^\ast(\alpha) \in (0,1)$ such that $\lim_{n\rightarrow \infty} Q^\alpha_n\Bigg(\frac{|S_n|}{n} \in \lbrack\theta^\ast(\alpha) - \varepsilon,\theta^\ast(\alpha) + \varepsilon\rbrack\Bigg) = 1 \text{for every} \varepsilon > 0.$ We give a characterization of $\theta^\ast(\alpha)$ in terms of the largest eigenvalue of a one-parameter family of $\mathbb{N} \times \mathbb{N}$ matrices. This allows us to prove that $\theta^\ast(\alpha)$ is an analytic function of the strength $\alpha$ of the self-repellence. In addition to the speed we prove a limit law for the local times of the random walk. The techniques used enable us to treat more general forms of self-repellence involving multiple intersections. We formulate a partial differential inequality that is equivalent to $\alpha \rightarrow \theta^\ast(\alpha)$ being (strictly) increasing. The verification of this inequality remains open.

Let $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ be i.i.d. with the uniform distribution on $(\lbrack 0,1\rbrack^2, \| \|)$, where $\| \|$ denotes the Euclidean norm. Using a new presentation of the Ajtai-Komlos-Tusnady (AKT) transportation algorithm, it is shown that the square exponential transportation cost $\inf_\pi \sum^n_{i=1} \exp\Bigg(\frac{\|X_i - Y_{\pi(i)}\|}{K(\log n/n)^{1/2}}\Bigg)^2,$ where $\pi$ ranges over all permutations of the integers $1,\ldots,n$, satisfies an integrability condition. This condition strengthens the optimal matching results of AKT and supports a recent conjecture of Talagrand. Rates of growth for the $L_p$ transportation cost are also found.

De Gunst has formulated a stochastic model for the growth of a certain type of plant cell population that initially consists of $n$ cells. The total cell number $N_n(t)$ as predicted by the model is a non-Markovian counting process. The relative growth of the population, $n^{-1}(N_n(t) - n)$, converges almost surely uniformly to a nonrandom function $X$. In the present paper we investigate the behavior of the limit process $X(t)$ as $t$ tends to infinity and determine the order of magnitude of the duration of the process $N_n(t)$. There are two possible causes for the process $N_n$ to stop growing, and correspondingly, the limit process $X(t)$ has a derivative $X'(t)$ that is the product of two factors, one or both of which may tend to zero as $t$ tends to infinity. It turns out that there is a remarkable discontinuity in the tail behavior of the processes. We find that if only one factor of $X'(t)$ tends to zero, then the rate at which the limit process reaches its final limit is much faster and the order of magnitude of the duration of the process $N_n$ is much smaller than when both occur approximately at the same time.

Martingale methods have played an important role in the theory of Galton-Watson processes and branching random walks. The (random) Fourier transform of the position of the particles in the $n$th generation, normalized by its mean, is a martingale. Under second moments assumptions on the branching this has been very useful to study the asymptotics of the branching random walk. Using a different normalization, we obtain a new martingale which is in $L^2$ under weak assumptions on the displacement of the particles and strong assumptions on the branching.

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be an i.i.d. family of positive random variables. For each set $\xi$ of vertices of $\mathbb{Z}^d$, its weight is defined as $S(\xi) = \sum_{\nu \in \xi}X_\nu$. A greedy lattice animal of size $n$ is a connected subset of $\mathbb{Z}^d$ of $n$ vertices, containing the origin, and whose weight is maximal among all such sets. Let $N_n$ denote this maximal weight. We show that if the expectation of $X^d_\nu(\log^+ X_\nu)^{d+ a}$ is finite for some $a > 0$, then w.p.1 $N_n \leq Mn$ eventually for some finite constant $M$. Estimates for the tail of the distribution of $N_n$ are also derived.

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables with $E\{X^d_0(\log^+ X_0)^{d+\varepsilon}\} < \infty$ for some $\varepsilon > 0$ and $d \geq 2$. Define $M_n$ and $N_n$ by \begin{align*} M_n &= \max\big\{\sum_{\nu \in \pi} X_\nu: \pi \text{a self-avoiding path of length}\quad n \\ \text{starting at the origin}\big\},\\ N_n &= \max\big\{\sum_{\nu\in \xi} X_\nu:\xi \quad\text{a lattice animal of size} \quad n \text{containing the origin}\big\}.\end{align*} Then it has been shown that there exist $M < \infty$ and $N < \infty$ such that $\frac{M_n}{n} \rightarrow M \quad\text{and}\quad \frac{N_n}{n} \rightarrow N \text{a.s. and in}\quad L^1$. In this paper we show that $M = N$ if and only if $X_0$ has bounded support and $P\{x_0 = R\} \geq p_c$, where $R$ is the right end point of support of $X_0$ and $p_c$ is the critical probability for site percolation on $\mathbb{Z}^d$.

The notion of the shape of a triangle can be used to define the shape of a $2 \times 2$ real matrix; we find that the shape of a matrix retains just the right amount of information required for determining the main features of the asymptotic behaviour, as $n\rightarrow\infty$, of $\mathbf{G}_n\mathbf{G}_{n-1}\cdots\mathbf{G}_1$, where the $\mathbf{G}_i$ are i.i.d. copies of a $2 \times 2$ random matrix $\mathbf{G}$. An alternative formula to the Furstenberg formula is proposed for the upper Lyapounov exponent of the probability distribution of $\mathbf{G}$. We find that in some cases, using our formula, the Lyapounov exponent is more susceptible to explicit calculation.

In papers about self-organizing data structures, it is often mentioned that the assumption of independence of successive requests of keys should be relaxed and that the dependence should assume the form of a locality phenomenon. In this setting, the move-to-front rule is considered to be of interest, but no optimality result concerning this rule has yet appeared. In this paper we assume that the sequence of required keys is a Markov chain with a transition kernel $P$ and we consider the class $\mathscr{F}^\ast$ of stochastic matrices $P$ such that move-to-front is optimal among on-line rules, with respect to the stationary search cost. We give properties of $\mathscr{F}^\ast$ that bear out the usual explanation of optimality of move-to-front by a locality phenomenon exhibited by the sequence of required keys. We explicitly produce a large subclass of $\mathscr{F}^\ast$, while showing that in some cases move-to-front is optimal with respect to the speed of convergence toward stationary search cost.

The study of orderings for positive dependence on bivariate empirical distributions can be viewed as the study of partial orderings on the set $S_N$ of all permutations of the integers $1,\ldots,N$. This paper extends earlier bivariate results to multivariate empirical distributions, with focus on the trivariate case. In terms of a newly defined notion of relative rearrangement, characterizations are given of the more positively upper orthant dependent ordering and related orderings. A new partial ordering describing concordance on $(S_N)^m$ is also introduced and connected with the positively upper orthant dependence ordering.