Celebratio Mathematica — Liggett — Complete Bibliography

Complete Bibliography

Works connected to Jeffrey Edward Steif

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T. M. Liggett and J. E. Steif: “Stochastic domination: The contact process, Ising models and FKG measures,” Ann. Inst. H. Poincaré Probab. Statist. 42 : 2 (March–April 2006), pp. 223–243. MR 2199800 Zbl 1087.60074 ArXiv math/0504530 article

We prove for the contact process on \( \mathbb{Z}^d \), 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 \( \mathbb{Z}^d \) dominate the same set of product measures. We show that this latter fact fails completely on the homogeneous 3-ary tree. We also provide a different distinction between \( \mathbb{Z}^d \) and the homogeneous 3-ary tree concerning stochastic domination and Ising models; while it is known that the plus states for different temperatures on \( \mathbb{Z}^d \) are never stochastically ordered, on the homogeneous 3-ary tree, almost the complete opposite is the case. Next, we show that on \( \mathbb{Z}^d \), 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.

T. M. Liggett, J. E. Steif, and B. Tóth: “Statistical mechanical systems on complete graphs, infinite exchangeability, finite extensions and a discrete finite moment problem,” Ann. Probab. 35 : 3 (2007), pp. 867–914. MR 2319710 Zbl 1126.44007 ArXiv math/0512191 article

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.

Jeffrey Edward Steif	Related
Bálint Tóth	Related