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Uspekhi Mat. Nauk, 2001, Volume 56, Issue 5(341), Pages 117–172 (Mi umn402)  

This article is cited in 3 scientific papers (total in 3 papers)

Gibbs and quantum discrete spaces

V. A. Malyshev

M. V. Lomonosov Moscow State University

Abstract: The Gibbs field is one of the central objects of modern probability theory, mathematical statistical physics, and Euclidean field theory. In this paper we introduce and study a natural generalization of this field to the case in which the background space (a lattice, a graph) on which the random field is defined is itself a random object. Moreover, this randomness is given neither a priori nor independent of the configuration; on the contrary, the space and the configuration on it depend on each other, and both objects are given by a Gibbs construction. We refer to the resulting distribution as a Gibbs family because it parametrizes Gibbs fields on different graphs belonging to the support of the distribution. We also study the quantum analogue of Gibbs families and discuss relationships with modern string theory and quantum gravity.

DOI: https://doi.org/10.4213/rm402

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English version:
Russian Mathematical Surveys, 2001, 56:5, 917–972

Bibliographic databases:

UDC: 519.219
MSC: Primary 60K35, 83C45, 82B20; Secondary 60G60, 82B10, 82B26, 57M15, 57R56, 81T30, 81T45, 0
Received: 17.11.2000

Citation: V. A. Malyshev, “Gibbs and quantum discrete spaces”, Uspekhi Mat. Nauk, 56:5(341) (2001), 117–172; Russian Math. Surveys, 56:5 (2001), 917–972

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. A. Krikun, “Boundaries of a random triangulation of a disk”, Discrete Math. Appl., 14:3 (2004), 301–315  mathnet  crossref  crossref  mathscinet  zmath
    2. Napolitano G.M., Turova T.S., “the Ising Model on the Random Planar Causal Triangulation: Bounds on the Critical Line and Magnetization Properties”, 162, no. 3, 2016, 739–760  crossref  mathscinet  zmath  isi  scopus  scopus
    3. Napolitano G.M., Turova T., “Geometric Random Graphs Vs Inhomogeneous Random Graphs”, Markov Process. Relat. Fields, 25:4, SI (2019), 615–637  isi
  • Успехи математических наук Russian Mathematical Surveys
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