Ted Cox - Academia.edu (original) (raw)
Papers by Ted Cox
Perplexing Problems in Probability, 1999
Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Her... more Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Here we describe recent results which show that, in two or more dimensions, it is also the limit of long range contact processes and long, short, and medium range voter models. Super-Brownian motion and its close relatives have arisen in a variety of different contexts in the last few years. In this article we focus on their appearance as limits of rescaled interacting particle systems and, in particular, on the results in Mueller and Tribe (1995), Durrett and Perkins (1998), and Cox, Durrett and Perkins (1998). Rather than providing complete proofs, which at present are still rather lengthy in some cases, we will focus on explaining why these theorems hold. These results are related in spirit, if not methodology, to recent work of Derbez and Slade (1998) on the convergence of "sufficiently spread out" rescaled lattice trees to Integrated Super-Excursion (ISE) in more than 8 dimensions, and to ongoing work of Derbez, van der Hofstad and Slade on convergence of sufficiently spread out oriented percolation to super-Brownian motion in more than 4 spatial dimensions. These results are described by Slade (1999) elsewhere in this volume. Super-Brownian motion arises as the limit of rescaled branching random walk and so is "trivial" in the mathematical physics terminology. In this same parlance one expects rescaled interacting systems to converge to this trivial limit above a critical dimension and at the critical dimension, perhaps with logarithmic corrections. For the particle systems we will consider in detail, this dimension is two, the critical dimension for recurrence of simple random walk or Brownian motion. We note that for the contact process this critical dimension (as opposed to d = 4 as in the above work on critical oriented percolation) arises because of our long range scaling in this setting. In order to describe super-Brownian motion we start with
Infection Control, 1985
The question of culturing "sore throats" of staff in community hospitals is an important one. 1 O... more The question of culturing "sore throats" of staff in community hospitals is an important one. 1 Obviously the protocol will impact on the well being of both staff and hospital. E.P. Bradley Hospital is a children's p s y c h i a t r i c h o s p i t a l w i t h 3 0 0 employees and a census averaging 56 patients. For many years employees have been cultured for the complaint of sore throat. The techniques are similar to that of Watanakunakom with the addition of microbiological confirmation. Between J u n e 1981 and April 1985, 1,377 cultures of staff were taken and 234 were positive for beta hemolytic streptococcus group A representing an incidence rate of 17%. This is a much larger recovery rate than that found in the previously mentioned study. The explanation of the difference may be the fact that our findings are from a child and adolescent hospital rather than a general hospital. We feel t h a t t h e c o n c l u s i o n s recorded in the Watanakunakom article are probably not generalizable.
Perplexing Problems in Probability, 1999
Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Her... more Super-Brownian motion was originally constructed as a scaling limit of branching random walk. Here we describe recent results which show that, in two or more dimensions, it is also the limit of long range contact processes and long, short, and medium range voter models. Super-Brownian motion and its close relatives have arisen in a variety of different contexts in the last few years. In this article we focus on their appearance as limits of rescaled interacting particle systems and, in particular, on the results in Mueller and Tribe (1995), Durrett and Perkins (1998), and Cox, Durrett and Perkins (1998). Rather than providing complete proofs, which at present are still rather lengthy in some cases, we will focus on explaining why these theorems hold. These results are related in spirit, if not methodology, to recent work of Derbez and Slade (1998) on the convergence of "sufficiently spread out" rescaled lattice trees to Integrated Super-Excursion (ISE) in more than 8 dimensions, and to ongoing work of Derbez, van der Hofstad and Slade on convergence of sufficiently spread out oriented percolation to super-Brownian motion in more than 4 spatial dimensions. These results are described by Slade (1999) elsewhere in this volume. Super-Brownian motion arises as the limit of rescaled branching random walk and so is "trivial" in the mathematical physics terminology. In this same parlance one expects rescaled interacting systems to converge to this trivial limit above a critical dimension and at the critical dimension, perhaps with logarithmic corrections. For the particle systems we will consider in detail, this dimension is two, the critical dimension for recurrence of simple random walk or Brownian motion. We note that for the contact process this critical dimension (as opposed to d = 4 as in the above work on critical oriented percolation) arises because of our long range scaling in this setting. In order to describe super-Brownian motion we start with
Infection Control, 1985
The question of culturing "sore throats" of staff in community hospitals is an important one. 1 O... more The question of culturing "sore throats" of staff in community hospitals is an important one. 1 Obviously the protocol will impact on the well being of both staff and hospital. E.P. Bradley Hospital is a children's p s y c h i a t r i c h o s p i t a l w i t h 3 0 0 employees and a census averaging 56 patients. For many years employees have been cultured for the complaint of sore throat. The techniques are similar to that of Watanakunakom with the addition of microbiological confirmation. Between J u n e 1981 and April 1985, 1,377 cultures of staff were taken and 234 were positive for beta hemolytic streptococcus group A representing an incidence rate of 17%. This is a much larger recovery rate than that found in the previously mentioned study. The explanation of the difference may be the fact that our findings are from a child and adolescent hospital rather than a general hospital. We feel t h a t t h e c o n c l u s i o n s recorded in the Watanakunakom article are probably not generalizable.