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Invariance, Quasi-Invariance, and Unimodularity for Random Graphs

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We treat 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 a very general dynamical scheme of constructing and studying measures with a prescribed Radon–Nikodym cocycle.

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References

  1. D. Aldous and R. Lyons, “Processes on unimodular random networks,” Electron. J. Probab., 12, 1454–1508 (2007) (electronic).

  2. D. Aldous and J. M. Steele, “The objective method: probabilistic combinatorial optimization and local weak convergence,” in: Probability on Discrete Structures, Encyclopaedia Math. Sci., 110, Springer, Berlin (2004), pp. 1–72.

  3. M. Aizenman and S. Warzel, “The canopy graph and level statistics for random operators on trees,” Math. Phys. Anal. Geom., 9, 291–333 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  4. I. Benjamini and N. Curien, “Ergodic theory on stationary random graphs,” Electron. J. Probab., 17, No. 93, 20 pp. (2012) (electronic).

  5. H. Bass and A. Lubotzky, Tree Lattices, Progress in Mathematics, 176, With appendices by H. Bass, L. Carbone, Lubotzky, G. Rosenberg, and J. Tits, Birkhäuser Boston, Inc., Boston, MA (2001).

  6. I. Benjamini, R. Lyons, Y. Peres, and O. Schramm, “Group-invariant percolation on graphs,” Geom. Funct. Anal., 9, 29–66 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  7. I. Benjamini, R. Lyons, and O. Schramm, “Unimodular random trees,” Ergodic Theory Dynam. Systems, 35, 359–373 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Bowen, “Equilibrium states and the ergodic theory of Anosov diffeomorphisms,” Lect. Notes Math., 470, Springer, Berlin–New York (1975).

  9. I. Benjamini and O. Schramm, “Recurrence of distributional limits of finite planar graphs,” Electron. J. Probab., 6, No. 23, 13 pp. (2001) (electronic).

  10. L. Carbone, Non-uniform Lattices on Uniform Trees, Mem. Amer. Math. Soc., 152, No. 724, xii+127 (2001).

  11. D. I. Cartwright, V. A. Kaimanovich, and W. Woess, “Random walks on the affine group of local fields and of homogeneous trees,” Ann. Inst. Fourier (Grenoble), 44, 1243–1288 (1994).

  12. L. Carbone and G. Rosenberg, “Lattices on nonuniform trees,” Geom. Dedicata, 98, 161–188 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Feldman and C. C. Moore, “Ergodic equivalence relations, cohomology, and von Neumann algebras. I,” Trans. Amer. Math. Soc., 234, 289–324 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  14. L. Garnett, “Foliations, the ergodic theorem and Brownian motion,” J. Funct. Anal., 51, 285–311 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  15. O. Häggström, “Infinite clusters in dependent automorphism invariant percolation on trees,” Ann. Probab., 25, 1423–1436 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  16. F. Harary, “Methods of destroying the symmetries of a graph,” Bull. Malays. Math. Sci. Soc. (2), 24, 183–191 (2001).

  17. F. Harary and D. Ranjan, “Breaking symmetry in complete graphs by orienting edges: asymptotic bounds,” Inform. Process. Lett., 67, 227–230 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  18. V. A. Kaimanovich, “Brownian motion on foliations: entropy, invariant measures, mixing,” Funkts. Anal. Prilozhen., 22, 82–83 (1988).

    MathSciNet  Google Scholar 

  19. V. A. Kaimanovich, “Measure-theoretic boundaries of Markov chains, 0-2 laws, and entropy,” in: Harmonic Analysis and Discrete Potential Theory (Frascati, 1991), Plenum, New York (1992), pp. 145–180.

  20. V. A. Kaimanovich, “Hausdorff dimension of the harmonic measure on trees,” Ergodic Theory Dynam. Systems, 18, 631–660 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. A. Kaimanovich, “Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization,” in: Fractals in Graz 2001, Trends Math., Birkhäuser, Basel (2003), pp. 145–183.

  22. V. A. Kaimanovich, “Amenability and the Liouville property,” Israel J. Math., 149, 45–85 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  23. V. A. Kaimanovich, “Hopf decomposition and horospheric limit sets,” Ann. Acad. Sci. Fenn. Math., 35, 335–350 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  24. V. A. Kaimanovich and M. Lyubich, Conformal and Harmonic Measures on Laminations Associated With Rational Maps, Mem. Amer. Math. Soc., 173, No. 820, vi+119 (2005).

  25. V. A. Kaimanovich and W. Woess, “Construction of discrete, nonunimodular hypergroups,” in: Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), World Sci. Publ., River Edge, NJ (1995) pp. 196–209.

  26. V. A. Kaimanovich and W. Woess, “Boundary and entropy of space homogeneous Markov chains,” Ann. Probab., 30, 323–363 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Lyons, R. Pemantle, and Yu. Peres, “Ergodic theory on Galton–Watson trees: speed of random walk and dimension of harmonic measure,” Ergodic Theory Dynam. Systems, 15, 593–619 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  28. J. R. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge Univ. Press, Cambridge (1998) (Reprint of the 1997 original).

  29. F. Paulin, “Propriétés asymptotiques des relations d’équivalences mesurées discrétes,” Markov Process. Related Fields, 5, 163–200 (1999).

    MathSciNet  Google Scholar 

  30. M. Rosenblatt, Markov Processes. Structure and Asymptotic Behavior, Springer–Verlag, New York (1971).

  31. M. A. Ronan and J. Tits, “Twin trees. II. Local structure and a universal construction,” Israel J. Math., 109, 349–377 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Schlichting, “Polynomidentitäten und Permutationsdarstellungen lokalkompakter Gruppen,” Invent. Math., 55, 97–106 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  33. Ya. G. Sinai, “Gibbs measures in ergodic theory,” Usp. Mat. Nauk, 27, No. 4 (166), 21–64 (1972).

  34. D. Sullivan, “The density at infinity of a discrete group of hyperbolic motions,” Inst. Hautes Études Sci. Publ. Math., No. 50, 171–202 (1979).

  35. P. M. Soardi and W. Woess, “Amenability, unimodularity, and the spectral radius of random walks on infinite graphs,” Math. Z., 205, 471–486 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  36. V. I. Trofimov, “Groups of automorphisms of graphs as topological groups,” Mat. Zametki, 38, 378–385 (1985).

    MathSciNet  Google Scholar 

  37. W. T. Tutte, “A census of planar triangulations,” Canad. J. Math., 14, 21–38 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  38. W. T. Tutte, “On the enumeration of convex polyhedra,” J. Combin. Theory Ser. B, 28, 105–126 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  39. W. Woess, “Topological groups and recurrence of quasitransitive graphs,” Rend. Sem. Mat. Fis. Milano, 64, 185–213 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Woess, Random Walks on Infinite Graphs and Groups, Cambridge Tracts in Mathematics, 138, Cambridge Univ. Press, Cambridge (2000).

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Authors and Affiliations

  1. University of Ottawa, Ottawa, ON, Canada

    V. A. Kaimanovich

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  1. V. A. Kaimanovich
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Correspondence to V. A. Kaimanovich.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 441, 2015, pp. 210–238.

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Kaimanovich, V.A. Invariance, Quasi-Invariance, and Unimodularity for Random Graphs. J Math Sci 219, 747–764 (2016). https://doi.org/10.1007/s10958-016-3144-z

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