Two-dimensional volume-frozen percolation: exceptional scales

We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when they reach diameter at least N was studied in earlier papers. Using volume as a way to measure the size of a cluster - instead of diameter - leads, for large N, to a quite different behavior (contrary to what happens on the binary tree, where the volume model and the diameter model are "asymptotically the same"). In particular, we show the existence of a sequence of "exceptional" length scales.Comment: 20 pages, 2 figure

Near-critical percolation with heavy-tailed impurities, forest fires and frozen percolation

Consider critical site percolation on a "nice" planar lattice: each vertex is occupied with probability $p = p_c$, and vacant with probability $1 - p_c$. Now, suppose that additional vacancies ("holes", or "impurities") are created, independently, with some small probability, i.e. the parameter $p_c$ is replaced by $p_c - \varepsilon$, for some small $\varepsilon > 0$. A celebrated result by Kesten says, informally speaking, that on scales below the characteristic length $L(p_c - \varepsilon)$, the connection probabilities remain of the same order as before. We prove a substantial and subtle generalization to the case where the impurities are not only microscopic, but allowed to be "mesoscopic". This generalization, which is also interesting in itself, was motivated by our study of models of forest fires (or epidemics). In these models, all vertices are initially vacant, and then become occupied at rate $1$. If an occupied vertex is hit by lightning, which occurs at a (typically very small) rate $\zeta$, its entire occupied cluster burns immediately, so that all its vertices become vacant. Our results for percolation with impurities turn out to be crucial for analyzing the behavior of these forest fire models near and beyond the critical time (i.e. the time after which, in a forest without fires, an infinite cluster of trees emerges). In particular, we prove (so far, for the case when burnt trees do not recover) the existence of a sequence of "exceptional scales" (functions of $\zeta$). For forests on boxes with such side lengths, the impact of fires does not vanish in the limit as $\zeta \searrow 0$.Comment: 67 pages, 15 figures (some small corrections and improvements, one additional figure); version to be submitte

A percolation process on the binary tree where large finite clusters are frozen

We study a percolation process on the planted binary tree, where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the process converges in some sense to the frozen percolation process introduced by Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.Comment: 11 page

On the four-arm exponent for 2D percolation at criticality

For two-dimensional percolation at criticality, we discuss the inequality $\alpha_4 > 1$ for the polychromatic four-arm exponent (and stronger versions, the strongest so far being $\alpha_4 \geq 1 + \frac{\alpha_2}{2}$, where $\alpha_2$ denotes the two-arm exponent). We first briefly discuss five proofs (some of them implicit and not self-contained) from the literature. Then we observe that, by combining two of them, one gets a completely self-contained (and yet quite short) proof.Comment: 23 pages, 3 figures; in memory of Vladas Sidoraviciu

Cysticercosis and taeniasis cases diagnosed at two referral medical institutions, Belgium, 1990 to 2015

Background: Few case reports on human infections with the beef tapeworm Taenia saginata and the pork tapeworm, Taenia solium, diagnosed in Belgium have been published, yet the grey literature suggests a higher number of cases. Aim: To identify and describe cases of taeniasis and cysticercosis diagnosed at two Belgian referral medical institutions from 1990 to 2015. Methods: In this observational study we retrospectively gathered data on taeniasis and cysticercosis cases by screening laboratory, medical record databases as well a uniform hospital discharge dataset. Results: A total of 221 confirmed taeniasis cases were identified. All cases for whom the causative species could be determined (170/221, 76.9%) were found to be T. saginata infections. Of those with available information, 40.0% were asymptomatic (26/65), 15.4% reported diarrhoea (10/65), 9.2% reported anal discomfort (6/65) and 15.7% acquired the infection in Belgium (11/70). Five definitive and six probable cases of neurocysticercosis (NCC), and two cases of non-central nervous system cysticercosis (non-CNS CC) were identified. Common symptoms and signs in five of the definitive and probable NCC cases were epilepsy, headaches and/or other neurological disorders. Travel information was available for of the 13 NCC and non-CNS CC cases; two were Belgians travelling to and eight were immigrants or visitors travelling from endemic areas. Conclusions: The current study indicates that a non-negligible number of taeniasis cases visit Belgian medical facilities, and that cysticercosis is occasionally diagnosed in international travellers

Generalizing about trade show effectiveness: a cross-national comparison.

Trade shows are a multi-billion dollar business in the US and the UK, but little is known about the determinants of trade show effectiveness. In this paper, we build a model that explains differences in trade show effectiveness across industries, across companies and across two countries. We focus on the differences in trade show effectiveness measured in a similar way across similar samples of 171 US and 135 UK firm-show experiences between 1980 and 1991. While the similarities outweigh the differences, we find evidence that trade shows are viewed differently by exhibitors and attendees in these two countries. We are able to make substantial generalizations about the effect of various show selection (go-not go) variables (booth size, personnel, etc.) on observed performance. We discuss the implications of our research for developing benchmarks for trade show performance and for better global management of the business marketing communications mix.Effectiveness; Trade;

Boundary rules and breaking of self-organized criticality in 2D frozen percolation

We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (“freeze”) as soon as their “size” becomes at least N, for some parameter N ≥ 1. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites). Diameter-frozen and volume-frozen percolation have been studied in previous works ([5, 15] and [6, 4], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these “boundary rules” in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case

Boundary rules and breaking of self-organized criticality in 2D frozen percolation

We study frozen percolation on the (planar) triangular lattice, where connected components stop growing ("freeze") as soon as their "size" becomes at least N, for some parameter N ≥ 1. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites). Diameter-frozen and volume-frozen percolation have been studied in previous works ([25, 11] and [27, 26], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these "boundary conditions" in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case