Nonlocal lattice fermionic models on a two-dimensional torus

Abelian fermionic models described by the SLAC action on a two-dimensional finite lattice are considered. In vector U(1) models modified by introducing additional Pauli-Villars regularization, nonlocal effects are suppressed, and the results are in good agreement with the continuous-theory results. For chiral fermions, the lattice determinant phase differs from the determinant phase in the continuous theory. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115, No. 1, pp. 93–105, April, 1998     

Scattering theory for a class of fermionic Pauli-Fierz models

The scattering theory for a class of fermionic Pauli-Fierz models is considered. We give a proof of the asymptotic completeness of the dynamics in the case of massive fermions. The result applied to the Hamiltonian of a quantized spin- Dirac particle interacting with an external field through a cutoff Yukawa interaction and to the Hamiltonian of a system of finitely many confined particles coupled to a fermionic field with a quadratic interaction.     

Asymptotics of partition functions in a fermionic matrix model and of related q‐polynomials

In this paper, we study asymptotics of the thermal partition function of a model of quantum mechanical fermions with matrix‐like index structure and quartic interactions. This partition function is given explicitly by a Wronskian of the Stieltjes‐Wigert polynomials. Our asymptotic results involve the theta function and its derivatives. We also develop a new asymptotic method for general q‐polynomials.     

M-theory of matrix models

Small M-theories incorporate various models representing a unified family in the same way that the M-theory incorporates a variety of superstring models. We consider this idea applied to the family of eigenvalue matrix models: their M-theory unifies various branches of the Hermitian matrix model (including the Dijkgraaf-Vafa partition functions) with the Kontsevich τ-function. Moreover, the corresponding duality relations are reminiscent of instanton and meron decompositions, familiar from the Yang-Mills theory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 2, pp. 179–192, February, 2007.     

Unitary integrals and related matrix models

We briefly review the basic properties of unitary matrix integrals, using three matrix models to analyze their properties: the ordinary unitary, the Brezin—Gross—Witten, and the Harish—Chandra—Itzykson—Zuber models. We especially emphasize the nontrivial aspects of the theory, from the De Witt’Hooft anomaly in unitary integrals to the problem of calculating correlators with the Itzykson-Zuber measure. We emphasize the method of character expansions as a technical tool. Unitary integrals are still insufficiently investigated, and many new results should be expected as this field attracts increased attention.     

Stochastic models of consumer behaviour

During the past thirty years much research has been carried out in order to produce decision aids for managers. Considerable efforts have been dedicated to models that describe how consumers make purchase decisions. Within this vast area of research we review stochastic approaches to consumer behaviour, i.e. probabilistic laws relating the observed strings of consecutive purchases to explanator variables. We start out by discussing the basic concepts underlying this field. Then we deal with models concentrating on brand choice, whereby we trace the development from the early ideas in the Fifties to today's sophisticated models. Subsequently we turn to the Negative Binomial Distribution, the ‘classical’ purchase incidence model, and its extensions. Then we describe combined purchase timing-brand selection models and introduce models of store choice and purchase quantity selection. After briefing on estimation and validation methods for stochastic models we finally critically review the state-of-the-art in this field. Special emphasis is placed on empirical applications throughout the review.     

Gauge theories as matrix models

We discuss the relation between the Seiberg-Witten prepotentials, Nekrasov functions, and matrix models. On the semiclassical level, we show that the matrix models of Eguchi-Yang type are described by instantonic contributions to the deformed partition functions of supersymmetric gauge theories. We study the constructed explicit exact solution of the four-dimensional conformal theory in detail and also discuss some aspects of its relation to the recently proposed logarithmic beta-ensembles. We also consider “quantizing” this picture in terms of two-dimensional conformal theory with extended symmetry and stress its difference from the well-known picture of the perturbative expansion in matrix models. Instead, the representation of Nekrasov functions using conformal blocks or Whittaker vectors provides a nontrivial relation to Teichmüller spaces and quantum integrable systems.     

Zero asymptotic behaviour for orthogonal matrix polynomials

Weak-star asymptotic results are obtained for the zeros of orthogonal matrix polynomials (i.e., the zeros of their determinants) on ℝ from two different assumptions: first from the convergence of matrix coefficients occurring in the three-term recurrence for these polynomials; and, second, from conditions on the generating matrix measure. The matrix analogues of the Chebyshev polynomials of the first kind are also investigated. The research of the first and second authors has been supported by DGICYT ref. PB96-1321-C02-01, and the research of the third author was supported, in part, by the U.S. National Science Foundation under the grant DMS-9501130.     

Critical exponents in fermionic hierarchical model

We consider the problem of algebraic computation of the critical exponent ν in the 2N-component fermionic Dyson model on a hierarchical lattice without the use of perturbation theory. Analyzing the results in a particular case when N = 2, we conclude that an algebraic approach in this model gives the same expression for ν as the approach of functional integration via Feynman diagrams in the p-adic ϕ ⁴-model.     

Identifying models of environmental systems' behaviour

This paper addresses the question of how theories are developed about the behaviour of large, complex systems such as those typically encountered in managing environmental quality. The specific problem considered is that of model structure identification by reference to experimental, in situ field data. A conceptual definition of this problem is given in terms of the notion of testing model hypotheses to the point of failure. An approach to solving the problem is proposed in which the use of recursive model parameter estimation algorithms is a central feature. This approach is illustrated by a case study in developing a dynamic model of water quality in the Bedford Ouse River in central-eastern England. The results are organized around the two principles of attempting to falsify confident hypotheses and of speculating about relatively uncertain hypotheses in order to modify inadequate prior hypotheses. The essential difficulty demonstrated by the case study is one of absorbing and interpreting the diagnostic evidence of field data analysis and this is ultimately a difficulty associated with the complex and intrinsically indivisible nature of large-scale systems.     

Topological symmetry,background independence and matrix models

We illustrate a physical situation in which topological symmetry, its breakdown, space-time uncertainty principle, and background independence may play an important role in constructing and understanding matrix models. First, we show that the space-time uncertainty principle of string may be understood as a manifestation of the breakdown of the topological symmetry in the large N matrix model. Next, we construct a new type of matrix models which is a matrix model analog of the topological Chern-Simons and BF theories. It is of interest that these topological matrix models are not only completely independent of the background metric but also have nontrivial “p-brane” solutions as well as commuting classical space-time as the classical solutions. In this paper, we would like to point out some elementary and unsolved problems associated to the matrix models, whose resolution would lead to the more satisfying matrix model in future.     

Continuum limit in the fermionic hierarchical model

We discuss the problem of rigorously constructing the continuum limit in the fermionic hierarchical model. The continuum limit constructed as the limit of fields on the refined hierarchical lattices is a field on a p-adic continuum. We investigate the problem of reconstructing the coupling constants of the continuum model from the coupling constants of the discretized model. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 40–50, January, 1999.     

Renormalization group in a fermionic hierarchical model

A study is made of a hierarchical model with spin values in a Grassmann algebra defined by a potential of general form. The action of the spin-block renormalization group in the space of Hamiltonians is reduced to a rational mapping of the space of coupling constants into itself. The methods of the theory of bifurcations are used to investigate the nontrivial fixed points of this mapping. A theorem establishing the existence of a thermodynamic limit of the model at these points in a certain neighborhood of a bifurcation value is proved.This work was done with financial support of the Russian Foundation for Fundamental Research (Grant 93-011-16099).State University, Kazan. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 282–293, November, 1994.