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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 2(356), Pages 65–104 (Mi umn718)  

This article is cited in 42 scientific papers (total in 43 papers)

Random metric spaces and universality

A. M. Vershik

St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences

Abstract: The notion of random metric space is defined, and it is proved that such a space is isometric to the Urysohn universal metric space with probability one. The main technique is the study of universal and random distance matrices; properties of metric (in particular, universal) spaces are related to properties of distance matrices. Examples of other categories in which randomness and universality coincide (graphs, and so on) are given.


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English version:
Russian Mathematical Surveys, 2004, 59:2, 259–295

Bibliographic databases:

UDC: 519.212.2+515.124.4
MSC: Primary 54E35, 54C25, 54E70; Secondary 28D99, 05C80, 15A50, 54E20, 60D05, 60B05
Received: 09.12.2003

Citation: A. M. Vershik, “Random metric spaces and universality”, Uspekhi Mat. Nauk, 59:2(356) (2004), 65–104; Russian Math. Surveys, 59:2 (2004), 259–295

Citation in format AMSBIB
\by A.~M.~Vershik
\paper Random metric spaces and universality
\jour Uspekhi Mat. Nauk
\yr 2004
\vol 59
\issue 2(356)
\pages 65--104
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 2
\pages 259--295

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    This publication is cited in the following articles:
    1. Baldwin J.T., “The metamathematics of random graphs”, Ann. Pure Appl. Logic, 143:1-3 (2006), 20–28  crossref  mathscinet  zmath  isi  elib
    2. Cameron P.J., Vershik A.M., “Some isometry groups of the Urysohn space”, Ann. Pure Appl. Logic, 143:1-3 (2006), 70–78  crossref  mathscinet  zmath  isi  elib
    3. Gao Su, Shao Chuang, “Random generations of the countable random graph”, Ann. Pure Appl. Logic, 143:1-3 (2006), 79–86  crossref  mathscinet  zmath  isi  elib
    4. Pestov V., “The isometry group of the Urysohn space as a Lévy group”, Topology Appl., 154:10 (2007), 2173–2184  crossref  mathscinet  zmath  isi  elib
    5. Kechris A.S., Rosendal Ch., “Turbulence, amalgamation, and generic automorphisms of homogeneous structures”, Proc. Lond. Math. Soc. (3), 94:2 (2007), 302–350  crossref  mathscinet  zmath  isi
    6. Melleray J., Petrov F., Vershik A., “Linearly rigid metric spaces”, C. R. Math. Acad. Sci. Paris, 344:4 (2007), 235–240  crossref  mathscinet  zmath  isi  elib
    7. Bordenave Ch., “Eigenvalues of Euclidean random matrices”, Random Structures Algorithms, 34:4 (2008), 515–532  crossref  mathscinet  isi
    8. E. Bohomolny, O. Bohigas, C. Schmit, “Distance matrices and isometric embeddings”, Zhurn. matem. fiz., anal., geom., 4:1 (2008), 7–23  mathnet  mathscinet  zmath  elib
    9. Hubička J., Nešetřil J., “A finite presentation of the rational Urysohn space”, Topology Appl., 155:14 (2008), 1483–1492  crossref  mathscinet  zmath  isi
    10. Melleray J., “Some geometric and dynamical properties of the Urysohn space”, Topology Appl., 155:14 (2008), 1531–1560  crossref  mathscinet  zmath  isi  elib
    11. Pestov V.G., “A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space”, Topology Appl., 155:14 (2008), 1561–1575  crossref  mathscinet  zmath  isi  elib
    12. Uspenskij V.V., “On subgroups of minimal topological groups”, Topology Appl., 155:14 (2008), 1580–1606  crossref  mathscinet  zmath  isi  elib
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    14. Melleray J., Petrov F.V., Vershik A.M., “Linearly rigid metric spaces and the embedding problem”, Fund. Math., 199:2 (2008), 177–194  crossref  mathscinet  zmath  isi  elib
    15. Nguyen Van Thé Lionel, Sauer N.W., “The Urysohn sphere is oscillation stable”, Geom. Funct. Anal., 19:2 (2009), 536–557  crossref  mathscinet  zmath  isi  elib
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    28. Bordenave Ch., “On Euclidean Random Matrices in High Dimension”, Electron. Commun. Probab., 18 (2013), 1–8  crossref  mathscinet  isi
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