Geometric expansion of the log-partition function of the anisotropic Heisenberg model

We study the asymptotic expansion of the log-partition function of the anisotropic Heisenberg model in a bounded domain as this domain is dilated to infinity. Using the Ginibre's representation of the anisotropic Heisenberg model as a gas of interacting trajectories of a compound Poisson process we find all the non-decreasing terms of this expansion. They are given explicitly in terms of functional integrals. As the main technical tool we use the cluster expansion method.Comment: 38 page

Abstract cluster expansion with applications to statistical mechanical systems

We formulate a general setting for the cluster expansion method and we discuss sufficient criteria for its convergence. We apply the results to systems of classical and quantum particles with stable interactions

Competence raising through teaching of ESD environmental key topics implementing “project technology”

Vysokoškolské vzdělání v Arménii v současné době čelí tlaku vzdělávacích reforem, jakož i novým rozvojovým možnostem. Realizace potřebných změn souvisí s tím, že kromě znalostí, schopností, dovedností a hodnot jsou v procesu učení zdůrazňovány kompetence. Balíček reforem rovněž vyžaduje nové uvedení do globálních otázek životního prostředí ve vysokoškolském vzdělávání. V podmínkách současné ekologické krize je jedním z nejdůležitějších úkolů navrhnout nové přístupy ke vzdělávání pro udržitelný rozvoj ve vysokoškolském vzdělávání. Hlavní cílem tohoto typu vzdělávání je pak zvýšit kompetence a účast žáků v ochraně přírody a při rozhodování.Tzv. UNESCO Chair ve vzdělávání pro udržitelný rozvoj v Centru pro ekologicko-noosférických studií Národní akademie věd Arménské republiky ve spolupráci s Národním institutem vzdělávání arménského Ministerstva školství a vědy realizují řadu školení a seminářů pro pedagogy a studenty z pedagogických ústavů – cílem je zvýšit jejich kompetence v klíčových oblastech a tématech životního prostředí a v kontextu udržitelného rozvoje, a to prostřednictvím projektové metody výuky. Pro tato školení a semináře byly vybrány pedagogicky orientované univerzity a instituce odborného vzdělávání, kde se výuka vedla projektovou metodou s vybranými nejlepšími studenty a vyučujícími. Během těchto školení a seminářů (zabývajících se různými problémy v oblasti životního prostředí) bylo proškoleno 44 učitelů a studentů z 5 vysokých škol, 3 odborných škol, 5 středních škol, a 3 nevládních organizací působících ve 3 oblastech Arménské republiky. Byla provedena evaluace a nejlepší vyhodnocené projekty byly prezentovány na diskusích u kulatých stolů.Higher education in Armenia is currently facing the pressure of reform as well as tremendous opportunities for development. Implementation of these reforms is conditioned by the fact that competence is emphasized among knowledge, abilities, skills and values. The package of reforms also requires a new introduction to global environmental issues in higher education. In the conditions of the present ecological crisis one of the most important issues is to develop new approaches to ESD in higher education. The main problem of ESD is to raise the competence and participation of learners regarding environmental protection and resolution.The UNESCO Chair on “Education for Sustainable Development” at the Center for Ecological-Noosphere Studies of the National Academy of Sciences of the Republic of Armenia in cooperation with the National Institute of Education of the Armenian Ministry of Education and Science are carrying out a series of trainings and seminars for educators and students from pedagogical institutes to increase their competence on key environmental topics in the context of sustainable development implementing project method of teaching. Pedagogical-oriented universities and vocational training institutions have been chosen, where trainings and seminars have been carried out with the best students and the teaching staff using project method. In the result of the trainings and seminars 44 teachers and learners from 5 universities, 3 vocational colleges, 5 high schools, 3 NGOs were trained in 3 regions of the Republic of Armenia (with different environmental problems). Evaluations were carried out and the best tested projects were presented at round table discussions

Strong cluster properties and geometric expansion of the log-partition function of the Ginibre gas

We consider the geometric expansion of the log-partition function of a system of interacting Brownian loops in a bounded domain. This expansion generalizes the corresponding one for a classical gas due to one of the authors (see [10]). It can also be considered as a natural generalization of a famous result of Kac (in "Can one hear the shape of a drum", Amer. Math. Monthly 73 (1966), 1--23), where in the case of one loop only the expansion of the partition function is calculated as the length of the time interval, on which the loop is defined, tends to 0. For the proof of the geometric expansion we need a new type of decay property of the Ursell function which can be described in the following way: If one integrates the Ursell function over all configurations " x 0 + ¯ of loops which contain a fixed loop x 0 starting at 0 and at least one loop of ¯ which touches the outside of the centered ball with radius R, then the integral with respect to x 0 has power decay in R

Penrose-Stable Interactions in Classical Statistical Mechanics

Poghosyan S, Zessin HN. Penrose-Stable Interactions in Classical Statistical Mechanics. Annales Henri Poincaré . 2021.For a pair potential Phi in a general underlying space X satisfying some natural and sufficiently general conditions in the sense of Penrose (J Math Phys 4:1312, 1963) and Poghosyan and Ueltschi (J Math Phys 50:053509, 2009) together with a locally finite measure rho on X we define by means of the so-called Ursell kernel a function r which is shown to be the correlation function of a unique process G, the limiting Gibbs process for (Phi, rho) with empty boundary conditions. This process is exhibited as a Gibbs process in the sense of Dobrushin, Lanford and Ruelle for a class of pair potentials, which contains classical stable and hard-core potentials that are called Penrose potentials here. Particularly, a class of positive potentials is included. Finally, for some class of Penrose potentials, we show that G is the unique Gibbs process for Phi. We use the classical method of Kirkwood-Salsburg equations. A decisive role is played by a generalization of Ruelle's estimate for correlation functions

Cluster Representations of Classical and Quantum Processes

Poghosyan S, Zessin HN. Cluster Representations of Classical and Quantum Processes. Moscow Mathematical Journal. 2019;19(1):133-151.A cluster representation of projections onto bounded regions of limiting measures due to Minlos and Malyshev is developed under weaker conditions and in a general abstract way which covers not only classical but also permanental and determinantal processes

Cluster Representations of Classical and Quantum Processes

Poghosyan S, Zessin HN. Cluster Representations of Classical and Quantum Processes. Moscow Mathematical Journal. 2019;19(1):133-151.A cluster representation of projections onto bounded regions of limiting measures due to Minlos and Malyshev is developed under weaker conditions and in a general abstract way which covers not only classical but also permanental and determinantal processes