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 Zap. Nauchn. Sem. POMI, 2015, Volume 441, Pages 210–238 (Mi znsl6235)

Invariance, quasi-invariance and unimodularity for random graphs

V. A. Kaimanovich

Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa ON, K1N 6N5, Canada

Abstract: We interpret the probabilistic notion of unimodularity for measures on the space of rooted locally finite connected graphs in terms of the theory of measured equivalence relations. It turns out that the right framework for this consists in considering quasi-invariant (rather than just invariant) measures with respect to the root moving equivalence relation. We define a natural modular cocycle of this equivalence relation, and show that unimodular measures are precisely those quasi-invariant measures whose Radon–Nikodym cocycle coincides with the modular cocycle. This embeds the notion of unimodularity into the very general dynamical scheme of constructing and studying measures with a prescribed Radon–Nikodym cocycle.

Key words and phrases: random graph, space of rooted graphs, equivalence relation, unimodular measure, invariance, Radon–Nikodym cocycle.

 Funding Agency Grant Number Canada Research Chair Natural Sciences and Engineering Research Council of Canada (NSERC) FP7/2007-2013 European Research Council 257110-RAWG

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English version:
Journal of Mathematical Sciences (New York), 2016, 219:5, 747–764

Bibliographic databases:

UDC: 519

Citation: V. A. Kaimanovich, “Invariance, quasi-invariance and unimodularity for random graphs”, Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, POMI, St. Petersburg, 2015, 210–238; J. Math. Sci. (N. Y.), 219:5 (2016), 747–764

Citation in format AMSBIB
\Bibitem{Kai15} \by V.~A.~Kaimanovich \paper Invariance, quasi-invariance and unimodularity for random graphs \inbook Probability and statistics. Part~22 \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 441 \pages 210--238 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6235} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504507} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 219 \issue 5 \pages 747--764 \crossref{https://doi.org/10.1007/s10958-016-3144-z}