CiteULike: Author Liggett

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

Thomas Liggett — 2007-07-31T10:09:20-00:00

(30 Jul 2007)

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.

Stochastic Interacting Systems: Contact, Voter and Exclusion Processes (Grundlehren der mathematischen Wissenschaften)

Thomas Liggett — 2007-07-27T21:06:34-00:00

(15 November 1999)

Interactive Particle Systems is a branch of Probability Theory with close connections to Mathematical Physics and Mathematical Biology. In 1985, the author wrote a book (T. Liggett, Interacting Particle System, ISBN 3-540-96069) that treated the subject as it was at that time. The present book takes three of the most important models in the area, and traces advances in our understanding of them since 1985. In so doing, many of the most useful techniques in the field are explained and developed, so that they can be applied to other models and in other contexts. Extensive Notes and References sections discuss other work on these and related models. Readers are expected to be familiar with analysis and probability at the graduate level, but it is not assumed that they have mastered the material in the 1985 book. This book is intended for graduate students and researchers in Probability Theory, and in related areas of Mathematics, Biology and Physics.

Interacting Particle Systems (Classics in Mathematics)

Thomas Liggett — 2007-07-27T21:05:35-00:00

(22 December 2004)

From the reviews "[...] This book presents a complete treatment of a new class of random processes, which have been studied intensively during the last fifteen years. None of this material has ever appeared in book form before. …The high quality of this work, [...] , makes a fascinating subject and its open problem as accessible as possible. [...]" F.L. Spitzer in Mathematical Reviews, 1986 "[...] However, it can be said that the author has succeeded in what even experts are seldom able to achieve: To write a clearcut and inspiring book on his favorite subject which meets most, if not all requirements which can be imposed on a comprehensive text on an important new field. The author can be congratulated on his excellent presentation of the theory of interacting particle systems. The book is highly recommended to everyone who works on or is interested in this subject: to probabilists, physicists and theoretical biologists. [...]" G. Rosenkranz in Methods of Information in Medicine, 1986

Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model

Richard Holley — 2007-03-10T22:59:40-00:00

The Annals of Probability, Vol. 3, No. 4. (1975), pp. 643-663.

A theorem exhibiting the duality between certain infinite systems of interacting stochastic processes and a type of branching process is proved. This duality is then used to study the ergodic properties of the infinite system. In the case of the vector model a complete understanding of the ergodic behavior is obtained.

A polymorphism within a conserved beta(1)-adrenergic receptor motif alters cardiac function and beta-blocker response in human heart failure.

SB Liggett — 2006-09-05T10:37:17-00:00

Proc Natl Acad Sci U S A, Vol. 103, No. 30. (25 July 2006), pp. 11288-11293.

Heterogeneity of heart failure (HF) phenotypes indicates contributions from underlying common polymorphisms. We considered polymorphisms in the beta(1)-adrenergic receptor (beta(1)AR), a beta-blocker target, as candidate pharmacogenomic loci. Transfected cells, genotyped human nonfailing and failing ventricles, and a clinical trial were used to ascertain phenotype and mechanism. In nonfailing and failing isolated ventricles, beta(1)-Arg-389 had respective 2.8 +/- 0.3- and 4.3 +/- 2.1-fold greater agonist-promoted contractility vs. beta(1)-Gly-389, defining enhanced physiologic coupling under relevant conditions of endogenous expression and HF. The beta-blocker bucindolol was an inverse agonist in failing Arg, but not Gly, ventricles, without partial agonist activity at either receptor; carvedilol was a genotype-independent neutral antagonist. In transfected cells, bucindolol antagonized agonist-stimulated cAMP, with a greater absolute decrease observed for Arg-389 (435 +/- 80 vs. 115 +/- 23 fmol per well). Potential pathophysiologic correlates were assessed in a placebo-controlled trial of bucindolol in 1,040 HF patients. No outcome was associated with genotype in the placebo group, indicating little impact on the natural course of HF. However, the Arg-389 homozygotes treated with bucindolol had an age-, sex-, and race-adjusted 38% reduction in mortality (P = 0.03) and 34% reduction in mortality or hospitalization (P = 0.004) vs. placebo. In contrast, Gly-389 carriers had no clinical response to bucindolol compared with placebo. Those with Arg-389 and high baseline norepinephrine levels trended toward improved survival, but no advantage with this allele and exaggerated sympatholysis was identified. We conclude that beta(1)AR-389 variation alters signaling in multiple models and affects the beta-blocker therapeutic response in HF and, thus, might be used to individualize treatment of the syndrome.

beta2-Adrenergic Receptor Polymorphisms and Sudden Cardiac Death: A Signal to Follow

Stephen Liggett — 2006-08-02T14:27:19-00:00

Circulation, Vol. 113, No. 15. (18 April 2006), pp. 1818-1820.

10.1161/CIRCULATIONAHA.105.618967

Complex haplotypes derived from noncoding polymorphisms of the intronless alpha2A-adrenergic gene diversify receptor expression

Kersten Small — 2006-06-14T09:08:30-00:00

PNAS, Vol. 103, No. 14. (4 April 2006), pp. 5472-5477.

alpha 2A-adrenergic receptors (alpha2AAR) regulate multiple central nervous system, cardiovascular, and metabolic processes including neurotransmitter release, platelet aggregation, blood pressure, insulin secretion, and lipolysis. Complex diseases associated with alpha2AAR dysfunction display familial clustering, phenotypic heterogeneity, and interindividual variability in response to therapy targeted to alpha2AARs, suggesting common, functional polymorphisms. In a multiethnic discovery cohort we identified 16 single-nucleotide polymorphisms (SNPs) in the alpha2AAR gene organized into 17 haplotypes of two major phylogenetic clades. In contrast to other adrenergic genes, variability of the alpha2AAR was primarily due to SNPs in the promoter, 5' UTR and 3' UTR, as opposed to the coding block. Marked ethnic variability in the frequency of SNPs and haplotypes was observed: one haplotype represented 70% of Caucasians, whereas Africans and Asians had a wide distribution of less common haplotypes, with the highest haplotype frequencies being 16% and 35%, respectively. Despite the compact nature of this intronless gene, local linkage disequilibrium between a number of SNPs was low and ethnic-dependent. Whole-gene transfections into BE(2)-C human neuronal cells using vectors containing the entire approx5.3-kb gene without exogenous promoters were used to ascertain the effects of haplotypes on alpha2AAR expression. Substantial differences (P < 0.001) in transcript and cell-surface protein expression, by as much as approx5-fold, was observed between haplotypes, including those with common frequencies. Thus, signaling by this virtually ubiquitous receptor is under major genetic influence, which may be the basis for highly divergent phenotypes in complex diseases such as systemic and pulmonary hypertension, heart failure, diabetes, and obesity. 10.1073/pnas.0601345103

Molecular mechanisms of beta2-adrenergic receptor function and regulation.

DW McGraw — 2006-02-20T09:26:56-00:00

Proc Am Thorac Soc, Vol. 2, No. 4. (2005)

It is now clear that the beta2-adrenergic receptor continuously oscillates between various conformations in the basal state, and that agonists act to stabilize one or more conformations. It is conceivable that synthetic agonists might be engineered to preferentially confine the receptor to certain conformations deemed clinically important while having a less stabilizing effect on unwanted conformations. In addition, studies of genetically engineered mice have revealed previously unrecognized cross-talk between the beta2-receptor and phospholipase C, such that removal of the primary dilating pathway results in downregulation of constrictive pathways and overactivity of the dilating pathway increases the contractile response. These results indicate a dynamic interaction between beta2-receptor activity and Gq-coupled receptors that constrict the airway. Potentially, then, during chronic beta-agonist therapy, expression of phospholipase C is increased, the functions of Gq-coupled constrictive receptors are enhanced, and there may be an increased tendency for clinical decompensation due to asthma and chronic obstructive pulmonary disease triggers. Antagonists to these receptors might be able to act synergistically with chronic beta-agonists to block the effect of phospholipase C. Alternatively, perhaps novel phospholipase C antagonists would provide the most efficacious approach to blocking the physiologic sequelae of cross-talk between the beta2-receptor and phospholipase C.

The Presence of Lys27 Instead of Asn27 in Human Phospholamban Promotes Sarcoplasmic Reticulum Ca2+-ATPase Superinhibition and Cardiac Remodeling.

Wen Zhao — 2006-02-17T09:38:11-00:00

Circulation (13 February 2006)

BACKGROUND: Phospholamban (PLN) is an inhibitor of the Ca(2+) affinity of sarcoplasmic reticulum (SR) Ca(2+)-ATPase (SERCA2). The amino acid sequence of PLN is highly conserved, and although all species contain asparagine (Asn), human PLN is unique in containing lysine (Lys) at amino acid 27. METHODS AND RESULTS: Human PLN was introduced in the null background. Expression of human PLN, at similar levels to mouse wild-type PLN, resulted in significant decreases in the affinity of SERCA2 for Ca(2+), attributed to unique spatial conformation of this PLN form and increases in its monomeric active unit compared with mouse PLN. The increased inhibition by human PLN was associated with attenuated cardiac contractility in the intact-animal, organ, and cardiomyocyte levels and with depressed calcium kinetics. These inhibitory effects could not be fully reversed even on maximal isoproterenol stimulation. There were no alterations in the expression levels of SERCA2, calsequestrin, ryanodine receptor, and FKBP12, although the sodium/calcium exchanger and the L-type Ca(2+) channel expression levels were upregulated. The depressed function resulted in increased heart/body weight ratios and phosphorylation levels of Akt, p38, and Erk1/2. CONCLUSIONS: Human PLN may play a more inhibitory role than that of other species in Ca(2+) cycling. Expression of human PLN in the mouse is compensated by alterations in Ca(2+)-handling proteins and cardiac remodeling in an effort to normalize cardiac contractility. Thus, the unique amino acid sequence of human PLN may be critical in maintaining a high cardiac reserve, which is of paramount importance in the regulation of human cardiac function.

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

Thomas Liggett — 2005-12-12T09:40:59-00:00

(9 Dec 2005)

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 includes all ferromagnetic Ising, Potts and 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 Ising model with an additional 3-body interaction. Finally, we study the question of how much the antiferromagnetic 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.

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

Thomas Liggett — 2005-04-27T15:11:34-00:00

(26 April 2005)

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 $λ$ 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.

How likely is an i.i.d. degree sequence to be graphical?

Richard Arratia — 2005-04-07T16:06:27-00:00

(6 April 2005)

Given i.i.d. positive integer valued random variables D_1,...,D_n, one can ask whether there is a simple graph on n vertices so that the degrees of the vertices are D_1,...,D_n. We give sufficient conditions on the distribution of D_i for the probability that this be the case to be asymptotically 0, 1/2 or strictly between 0 and 1/2. These conditions roughly correspond to whether the limit of nP(D_i≥ n) is infinite, zero or strictly positive and finite. This paper is motivated by the problem of modeling large communications networks by random graphs.