Stochastic domination: the contact process, Ising models and FKG measures

We prove for the contact process on $Z^d$, and many other graphs, that the upper invariant measure dominates a homogeneous product measure with large density if the infection rate $\lambda$ is sufficiently large. As a consequence, this measure percolates if the corresponding product measure percolates. We raise the question of whether domination holds in the symmetric case for all infinite graphs of bounded degree. We study some asymmetric examples which we feel shed some light on this question. We next obtain necessary and sufficient conditions for domination of a product measure for ``downward'' FKG measures. As a consequence of this general result, we show that the plus and minus states for the Ising model on $Z^d$ dominate the same set of product measures. We show that this latter fact fails completely on the homogenous 3-ary tree. We also provide a different distinction between $Z^d$ and the homogenous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on $Z^d$ are never stochastically ordered, on the homogenous 3-ary tree, almost the complete opposite is the case. Next, we show that on $Z^d$, the set of product measures which the plus state for the Ising model dominates is strictly increasing in the temperature. Finally, we obtain a necessary and sufficient condition for a finite number of variables, which are both FKG and exchangeable, to dominate a given product measure.Comment: 27 page

Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem

We show that a large collection of statistical mechanical systems with quadratically represented Hamiltonians on the complete graph can be extended to infinite exchangeable processes. This extends a known result for the ferromagnetic Curie--Weiss Ising model and includes as well all ferromagnetic Curie--Weiss Potts and Curie--Weiss Heisenberg models. By de Finetti's theorem, this is equivalent to showing that these probability measures can be expressed as averages of product measures. We provide examples showing that ``ferromagnetism'' is not however in itself sufficient and also study in some detail the Curie--Weiss Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic Curie--Weiss Ising model can be extended. In this direction, we obtain sharp asymptotic results via a solution to a new moment problem. We also obtain a ``formula'' for the extension which is valid in many cases.Comment: Published at http://dx.doi.org/10.1214/009117906000001033 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Microform Editions of Documentary Collections: Where Do We Stand? And Where Do We Go From Here?

The following essay is intended to serve as a brief introduction to the world of scholarly micro-publishing. It will begin with a discussion of commercial micro-publishers and the contributions they can make towards the publication of a high-quality microform edition. It will also allude to some recent developments in the micro-publication of documentary collections, such as comprehensive microfiche editions and microfiche supplements to printed books, computer-generated microfiche, and selective (rather than comprehensive) microform editions. The essay will not include a technical discussion of microforms or a step-by-step description of how to prepare a collection of documents for filming. Information about these topics can be found in the works cited in the appended bibliography

Rule of Law with Asian Characteristics: cultural insights from the Occupy Central Movement in Hong Kong

This article uses the Occupy Central social movement in Hong Kong as a natural experiment to consider whether Asian culture influences the understanding and exercise of fundamental rights. In an earlier article, the author explored the relationship between Chinese culture and the rule of law as measured by the World Justice Project Rule of Law Index. This article relies on the earlier work but expands the analysis to consider Asian culture while at the same time focusing specifically on the fundamental rights aspect of the rule of law index. This article shows a strong correlation between lower scores for Asian countries on Individualism, as measured by the Hofstede Dimensions of Culture, and scores on fundamental rights, protection for freedom expression, and protection for freedom of association, as measured by the World Justice Project Index. This correlation is reflected by the Occupy Central movement. Although the movement was an exercise in fundamental rights, its purpose was to promote the rights of the community and the protests were carried out in a manner to reduce the impact on community rights When the movement ended, the negative consequences for the community were a significant reason, and the leaders of the movement were ultimately prosecuted for inciting a public nuisance