Front propagation in certain one-dimensional exclusion models

Recent results on two interacting particle systems on are summarized, the asymmetric simple exclusion process and the branching exclusion process.     

Actinide bioassays by ICPMS

The ever-increasing sensitivity of ICPMS continues to expand the technique’s application in the field of health physics. Enhancements in sample introduction and instrument design over the last few years have resulted in improving the ICPMS detection limit from ∼10 ng/l to≤0.1 ng/l. This additional sensitivity provides greater flexibility in the analysis of long-lived radionuclides in biological fluids, and requires only minimal sample preparation of urine for uranium analysis; the described 3-minute abbreviated matrix separation provides detection limits that are comparable to or better than alpha counting. For urine samples tested having concentrations that exceed the accepted administrative limit for total uranium (0.2 μg/day), isotopic analysis by ICPMS (e.g., determining the presence of²³⁶U, or measuring appropriate uranium isotope ratios) provides a reliable indication of occupational exposure. Our laboratory also utilizes ICPMS in a study examining uranium dissolution rate classification of dust collected at the perimeter of a nuclear facility. Specific details regarding these and other health physics applications are featured, including our group’s participation in assisting the DOE with the evaluation of ICPMS as a cost-effective alternative to fission-track analysis for the routine determination of²³⁹Pu in urine.     

Stability of Earliest-Due-Date,First-Served Queueing Networks

We study multiclass queueing networks with the earliest-due-date, first-served (EDDFS) discipline. For these networks, the service priority of a customer is determined, upon its arrival in the network, by an assigned random due date. First-in-system, first-out queueing networks, where a customer's priority is given by its arrival time in the network, are a special case. Using fluid models, we show that EDDFS queueing networks, without preemption, are stable whenever the traffic intensity satisfies _j <1 for each station j.     

State space collapse with application to heavy traffic limits for multiclass queueing networks

Heavy traffic limits for multiclass queueing networks are a topic of continuing interest. Presently, the class of networks for which these limits have been rigorously derived is restricted. An important ingredient in such work is the demonstration of state space collapse. Here, we demonstrate state space collapse for two families of networks, first-in first-out (FIFO) queueing networks of Kelly type and head-of-the-line proportional processor sharing (HLPPS) queueing networks. We then apply our techniques to more general networks. To demonstrate state space collapse for FIFO networks of Kelly type and HLPPS networks, we employ law of large number estimates to show a form of compactness for appropriately scaled solutions. The limits of these solutions are next shown to satisfy fluid model equations corresponding to the above queueing networks. Results from Bramson [4,5] on the asymptotic behavior of these limits then imply state space collapse. The desired heavy traffic limits for FIFO networks of Kelly type and HLPPS networks follow from this and the general criteria set forth in the companion paper Williams [41]. State space collapse and the ensuing heavy traffic limits also hold for more general queueing networks, provided the solutions of their fluid model equations converge. Partial results are given for such networks, which include the static priority disciplines. This revised version was published online in June 2006 with corrections to the Cover Date.     

A simple proof of the stability criterion of Gray and Griffeath

Summary Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having “all 0's” and “all 1's” as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the “all 1's” equilibrium under small perturbations. In this paper we use a renormalized site construction to give a much simpler proof. Our new approach can be used in many situations as a substitute for the contour method. Partially supported by a grant from the National Science Foundation Partially supported by the Army Research Office through the Mathematical Sciences Institute at Cornell University     

Occupation time large deviations of the voter model

This paper is a sequel to [5] and [6]. We continue our study of occupation time large deviation probabilities for some simple infinite particle systems by analysing the so-called voter model _t (see e.g., [11] or [8]). In keeping with our previous results, we show that the large deviations are classical in high dimensions (d5 for _t) but fat in low dimensions (d4). Interaction distinguishes the voter model from the independent particle systems of [5] and [6], and consequently exact computations no longer seem feasible. Instead, we derive upper and lower bounds which capture the asymptotic decay rate of the large deviation tails.Dedicated to Frank Spitzer on his 60th birthdayPartially supported by the National Science Foundation under Grant DMS-831080Partially supported by the National Science Foundation under Grant DMS-841317Partially supported by the National Science Foundation under Grant DMS-830549     

The survival of branching annihilating random walk

Summary Branching annihilating random walk is an interacting particle system on . As time evolves, particles execute random walks and branch, and disappear when they meet other particles. It is shown here that starting from a finite number of particles, the system will survive with positive probability if the random walk rate is low enough relative to the branching rate, but will die out with probability one if the random walk rate is high. Since the branching annihilating random walk is non-attractive, standard techniques usually employed for interacting particle systems are not applicable. Instead, a modification of a contour argument by Gray and Griffeath is used.     

Minimal displacement of branching random walk

Summary Let denote a branching random walk in with mean particle productionm, m>1, and with incremental spatial distributionG, withG({0}) =p andG({1})=1–p. Ifmp=1, then the minimal displacement of behaves asymptotically like log logn/log 2. If the conditionG({1})=1–p is replaced byG((0, ))=1–p, we obtain a similar result.Research was partially supported by the National Science Foundation under grant MCS-7607039