T. S. Mountford

Critical length for semi-oriented bootstrap percolation

We consider the behaviour of semi-oriented bootstrap percolation restricted to a finite square or torus. We prove that as the probability of initial occupancy p tends to zero, the side length required for a two-dimensional torus to have non-negligible chance of filling itself up is between for universal constants c and C. We show similar results for the side length required for a square to show significant clustering behaviour.

Clustering in one-dimensional threshold voter models

We consider one-dimensional spin systems in which the transition rate is 1 at site k if there are at least N sites in {k-N, k-N + 1, ..., k + N-1, k + N} at which the opinion differs from that at k, and the rate is zero otherwise. We prove that clustering occurs for all N [greater-or-equal, slanted] 1 in the sense that P[[eta]t(k) [not equal to] [eta]t(j)] tends to zero as t tends to [infinity] for every initial configuration. Furthermore, the limiting distribution as t --> [infinity] exists (and is a mixture of the pointmasses on [eta] [reverse not equivalent] 1 and [eta] [reverse not equivalent] 0) if the initial distribution is translation invariant. In case N = 1, the first of these results was proved and a special case of the second was conjectured in a recent paper by Cox and Durrett. Now let D([varrho]) be the limiting density of 1s when the initial distribution is the product measure with density [rho]. If N = 1, we show that D([rho]) is concave on [0, ], convex on [, 1], and has derivative 2 at 0. If N [greater-or-equal, slanted] 2, this derivative is zero.

Rates for the probability of large cubes being non-internally spanned in modified bootstrap percolation

SummaryWe find the exact rate of decay for the probability that a large cube is not internally spanned for the modified bootstrap percolation. It is proven that for cubes of large side the event that the cube is not internally spanned is essentially the same as the event that the cube possesses a completely vacant line.

Jordan curves in the level sets of additive Brownian motion

Keywords: additive Brownian motion ; Brownian sheet ; level set ; Jordan curve Reference PROB-ARTICLE-2001-002doi:10.1090/S0002-9947-01-02811-2View record in Web of Science Record created on 2008-12-01, modified on 2017-05-12

An Extension of Kuczek's Argument to Nonnearest Neighbor Contact Processes

The right edge of a nearest neighbor supercritical contact process satisfies a central limit theorem.(8,xa09) In this paper, a block construction is created to extend the argument by Kuczek(9) to the nonnearest neighbor case. The proof of the following fact in the nonnearest neighbor case is the key to the extension: There is a positive chance that the rightmost particle at time 0 infects the rightmost particle at every time.

An extension of a result of Burdzy and Lawler

SummaryIt is shown that for all mean zero, finite variance random walks, the critical non-intersection exponents are equal to those for Brownian motion. The method uses the local time of intersection.

THE SURVIVAL OF LARGE DIMENSIONAL THRESHOLD CONTACT PROCESSES

We study the threshold theta contact process on Z(d) with infection parameter lambda. We show that the critical point lambda(c), defined as the threshold for survival starting from every site occupied, vanishes as d -> infinity. This implies that the threshold theta voter model on Z(d) has a nondegenerate extremal invariant measure, when d is large.

Non-independence of excursions of the Brownian sheet and of additive Brownian motion

Keywords: Brownian sheet ; excursions ; level sets ; additive ; Brownian motion Reference PROB-ARTICLE-2003-001doi:10.1090/S0002-9947-02-03138-0View record in Web of Science Record created on 2008-12-01, modified on 2017-05-12

The ergodicity of a class of reversible reaction-diffusion processes

SummaryWe build on recent results of Durrett, Ding and Liggett to establish ergodicity in a class of reversible reaction-diffusion processes.

Limiting behaviour of the occupation of wedges by complex Brownian motion

SummaryWe prove a theorem which gives the lim inf behaviour ast tends to 0 for the amount of time a complex Brownian motion spends in a wedge with apex at the origin. The result is then shown to hold uniformly for all wedges a.s..