Journal ArticleDOI
Towards a theory of negative dependence
TLDR
In this article, the authors present examples that motivate a need for such a theory, give plausibility arguments for the existence of a theory of positively dependent events, outline a few possible directions a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.Abstract:
The FKG theorem says that the positive lattice condition, an easily checkable hypothesis which holds for many natural families of events, implies positive association, a very useful property. Thus there is a natural and useful theory of positively dependent events. There is, as yet, no corresponding theory of negatively dependent events. There is, however, a need for such a theory. This paper, unfortunately, contains no substantial theorems. Its purpose is to present examples that motivate a need for such a theory, give plausibility arguments for the existence of such a theory, outline a few possible directions such a theory might take, and state a number of specific conjectures which pertain to the examples and to a wish list of theorems.read more
Citations
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Book
Probability on Trees and Networks
Russell Lyons,Yuval Peres +1 more
TL;DR: In this article, the authors present a state-of-the-art account of probability on networks, including percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks.
Book
The Random-Cluster Model
TL;DR: The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures and of electrical networks as mentioned in this paper.
Journal ArticleDOI
Determinantal probability measures
Russell Lyons,Russell Lyons +1 more
TL;DR: In this paper, the authors studied basic combinatorial and probabilistic aspects in the discrete case of determinantal point processes, including relationships with matroids, stochastic domination, negative association, completeness for infinite matroid, tail triviality, and a method for extension of results from orthogonal projections to positive contractions.
Journal ArticleDOI
Uniform spanning forests
TL;DR: The uniform spanning forest model is related to random cluster models in statistical physics, but, because of the preceding connections, its analysis can be carried further as discussed by the authors, and several open problems and conjectures.
References
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Book ChapterDOI
Some Concepts of Dependence
TL;DR: In this article, the authors give three successively stronger definitions of positive dependence, and investigate their consequences, explore the strength of each definition through a number of examples, and give some statistical applications.
Journal ArticleDOI
Negative Association of Random Variables with Applications
Kumar Joag-Dev,Frank Proschan +1 more
TL;DR: In this article, negative association is defined as the property that a random variable is negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of Ω(1, 2, \cdots, k, k) of a function f(X, i, i = 1, 3, 4, k), f(G, g), g, g, j, j \in A_1), g(G), g), rbrack \leq 0, for all nondecreasing
Journal ArticleDOI
Association of Random Variables, with Applications
TL;DR: In this paper, it was shown that a random variable can be associated with another random variable if the test functions are either (a) binary or (b) bounded and continuous.
Journal ArticleDOI
Correlation inequalities on some partially ordered sets
TL;DR: In this article, it was shown that increasing functions on a finite distributive lattice are positively correlated by positive measures satisfying a suitable convexity property, and applications to Ising ferromagnets in an arbitrary magnetic field and to the random cluster model were given.
Journal ArticleDOI
A lower bound for the critical probability in a certain percolation process
TL;DR: In this paper, a random maze with the designations active or passive attached to the links is considered, where each link is assigned the designation active with probability p or passive with probability 1 − p, independently of all other links.