The single ring theorem

Abstract

We study the empirical measure $L_{A_n}$ of the eigenvalues of nonnormal square matrices of the form $A_n=U_nT_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $T_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $T_n$ converges, and $T_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure $\mu$ on the complex plane whose support is a single ring. In particular, we provide a complete proof of the Feinberg-Zee single ring theorem [FZ]. We also consider the case where $U_n,V_n$ are independently Haar distributed on the orthogonal group.

  • [AGZ] G. W. Anderson, A. Guionnet, and O. Zeitouni, An Introduction to Random Matrices, Cambridge: Cambridge Univ. Press, 2010, vol. 118.
    @book {AGZ, MRKEY = {2760897},
      AUTHOR = {Anderson, Greg W. and Guionnet, Alice and Zeitouni, Ofer},
      TITLE = {An Introduction to Random Matrices},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {118},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2010},
      PAGES = {xiv+492},
      ISBN = {978-0-521-19452-5},
      MRCLASS = {60B20 (46L53 46L54)},
      MRNUMBER = {2760897},
      ZBLNUMBER = {1184.15023},
      }
  • [bai] Go to document Z. D. Bai, "Circular law," Ann. Probab., vol. 25, iss. 1, pp. 494-529, 1997.
    @article {bai, MRKEY = {1428519},
      AUTHOR = {Bai, Z. D.},
      TITLE = {Circular law},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {25},
      YEAR = {1997},
      NUMBER = {1},
      PAGES = {494--529},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {60F15 (62H99)},
      MRNUMBER = {1428519},
      DOI = {10.1214/aop/1024404298},
      ZBLNUMBER = {0871.62018},
      }
  • [bl01] Go to document P. Biane and F. Lehner, "Computation of some examples of Brown’s spectral measure in free probability," Colloq. Math., vol. 90, iss. 2, pp. 181-211, 2001.
    @article {bl01, MRKEY = {1876844},
      AUTHOR = {Biane, Philippe and Lehner, Franz},
      TITLE = {Computation of some examples of {B}rown's spectral measure in free probability},
      JOURNAL = {Colloq. Math.},
      FJOURNAL = {Colloquium Mathematicum},
      VOLUME = {90},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {181--211},
      ISSN = {0010-1354},
      CODEN = {CQMAAQ},
      MRCLASS = {46L54},
      MRNUMBER = {1876844},
      MRREVIEWER = {Dimitri Y. Shlyakhtenko},
      DOI = {10.4064/cm90-2-3},
      ZBLNUMBER = {0988.22004},
      }
  • [brown] L. G. Brown, "Lidskiĭ’s theorem in the type ${ II}$ case," in Geometric Methods in Operator Algebras, Harlow: Longman Sci. Tech., 1986, vol. 123, pp. 1-35.
    @incollection {brown, MRKEY = {0866489},
      AUTHOR = {Brown, L. G.},
      TITLE = {Lidskiĭ's theorem in the type {${\rm II}$} case},
      BOOKTITLE = {Geometric Methods in Operator Algebras},
      VENUE={{K}yoto, 1983},
      SERIES = {Pitman Res. Notes Math. Ser.},
      VOLUME = {123},
      PAGES = {1--35},
      PUBLISHER = {Longman Sci. Tech.},
      ADDRESS = {Harlow},
      YEAR = {1986},
      MRCLASS = {47B10 (46L10)},
      MRNUMBER = {0866489},
      MRREVIEWER = {Thierry Fack},
      ZBLNUMBER = {0646.46058},
      }
  • [CGM] Go to document B. Collins, A. Guionnet, and E. Maurel-Segala, "Asymptotics of unitary and orthogonal matrix integrals," Adv. Math., vol. 222, iss. 1, pp. 172-215, 2009.
    @article {CGM, MRKEY = {2531371},
      AUTHOR = {Collins, Beno{\^ı}t and Guionnet, Alice and Maurel-Segala, Edouard},
      TITLE = {Asymptotics of unitary and orthogonal matrix integrals},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {222},
      YEAR = {2009},
      NUMBER = {1},
      PAGES = {172--215},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {46L54 (15A52 46L53 82B44)},
      MRNUMBER = {2531371},
      DOI = {10.1016/j.aim.2009.03.019},
      ZBLNUMBER = {1184.15024},
      }
  • [FZ] Go to document J. Feinberg and A. Zee, "Non-Gaussian non-Hermitian random matrix theory: phase transition and addition formalism," Nuclear Phys. B, vol. 501, iss. 3, pp. 643-669, 1997.
    @article {FZ, MRKEY = {1477381},
      AUTHOR = {Feinberg, Joshua and Zee, A.},
      TITLE = {Non-{G}aussian non-{H}ermitian random matrix theory: phase transition and addition formalism},
      JOURNAL = {Nuclear Phys. B},
      FJOURNAL = {Nuclear Physics. B},
      VOLUME = {501},
      YEAR = {1997},
      NUMBER = {3},
      PAGES = {643--669},
      ISSN = {0550-3213},
      CODEN = {NUPBBO},
      MRCLASS = {82B44 (15A52 60H25)},
      MRNUMBER = {1477381},
      MRREVIEWER = {Alexei Khorunzhy},
      DOI = {10.1016/S0550-3213(97)00419-7},
      ZBLNUMBER = {0933.82024},
      }
  • [FS] Y. V. Fyodorov and H. J. Sommers, "Spectra of random contractions and scattering theory for discrete-time systems," JETP Lett., vol. 72, pp. 422-426, 2000.
    @article{FS,
      author={Fyodorov, Yan V. and Sommers, H. J.},
      TITLE={Spectra of random contractions and scattering theory for discrete-time systems},
      JOURNAL={JETP Lett.},
      VOLUME={72},
      YEAR={2000},
      PAGES={422--426},
      }
  • [FW] Go to document Y. Wei and Y. V. Fyodorov, "On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values," J. Phys. A, vol. 41, iss. 50, p. 502001, 2008.
    @article {FW, MRKEY = {2515905},
      AUTHOR = {Wei, Yi and Fyodorov, Yan V. },
      TITLE = {On the mean density of complex eigenvalues for an ensemble of random matrices with prescribed singular values},
      JOURNAL = {J. Phys. A},
      FJOURNAL = {Journal of Physics. A. Mathematical and Theoretical},
      VOLUME = {41},
      YEAR = {2008},
      NUMBER = {50},
      PAGES = {502001},
      ISSN = {1751-8113},
      MRCLASS = {15B52 (15A16 15A18)},
      MRNUMBER = {2515905},
      MRREVIEWER = {Nizar Demni},
      DOI = {10.1088/1751-8113/41/50/502001},
      ZBLNUMBER = {1154.81343},
      }
  • [ginibre] Go to document J. Ginibre, "Statistical ensembles of complex, quaternion, and real matrices," J. Mathematical Phys., vol. 6, pp. 440-449, 1965.
    @article {ginibre, MRKEY = {0173726},
      AUTHOR = {Ginibre, Jean},
      TITLE = {Statistical ensembles of complex, quaternion, and real matrices},
      JOURNAL = {J. Mathematical Phys.},
      FJOURNAL = {Journal of Mathematical Physics},
      VOLUME = {6},
      YEAR = {1965},
      PAGES = {440--449},
      ISSN = {0022-2488},
      MRCLASS = {22.60 (53.90)},
      MRNUMBER = {0173726},
      MRREVIEWER = {J. Dieudonn{é}},
      DOI = {10.1063/1.1704292},
      ZBLNUMBER = {0127.39304},
      }
  • [girko] V. L. Girko, "The circular law," Teor. Veroyatnost. i Primenen., vol. 29, iss. 4, pp. 669-679, 1984.
    @article {girko, MRKEY = {0773436},
      AUTHOR = {Girko, V. L.},
      TITLE = {The circular law},
      JOURNAL = {Teor. Veroyatnost. i Primenen.},
      FJOURNAL = {Akademiya Nauk SSSR. Teoriya Veroyatnosteĭi ee Primeneniya},
      VOLUME = {29},
      YEAR = {1984},
      NUMBER = {4},
      PAGES = {669--679},
      ISSN = {0040-361X},
      MRCLASS = {15A52 (60F99)},
      MRNUMBER = {0773436},
      MRREVIEWER = {Ludwig Arnold},
      ZBLNUMBER = {0565.60034},
      }
  • [gotzetikhomirov] Go to document F. Götze and A. Tikhomirov, "The circular law for random matrices," Ann. Probab., vol. 38, iss. 4, pp. 1444-1491, 2010.
    @article {gotzetikhomirov, MRKEY = {2663633},
      AUTHOR = {G{ö}tze, Friedrich and Tikhomirov, Alexander},
      TITLE = {The circular law for random matrices},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {38},
      YEAR = {2010},
      NUMBER = {4},
      PAGES = {1444--1491},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {60B20},
      MRNUMBER = {2663633},
      DOI = {10.1214/09-AOP522},
      ZBLNUMBER = {1203.60010},
      }
  • [Guionnet] Go to document A. Guionnet, Large Random Matrices: Lectures on Macroscopic Asymptotics, New York: Springer-Verlag, 2009, vol. 1957.
    @book {Guionnet, MRKEY = {2498298},
      AUTHOR = {Guionnet, Alice},
      TITLE = {Large Random Matrices: Lectures on Macroscopic Asymptotics},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1957},
      NOTE = {Lectures from the 36th Probability Summer School held in Saint-Flour, 2006},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2009},
      PAGES = {xii+294},
      ISBN = {978-3-540-69896-8},
      MRCLASS = {60B20 (46L54 46N30 60E15 60H25 60H30)},
      MRNUMBER = {2498298},
      MRREVIEWER = {Vladislav Kargin},
      DOI = {10.1007/978-3-540-69897-5},
      ZBLNUMBER = {1168.60003},
      }
  • [haagerup] U. Haagerup, Spectral decomposition of all operators in a ${ II}_1$ factor, which is embeddable in ${R}^\omega$, 2001.
    @misc{haagerup,
      author={Haagerup, Uffe},
      TITLE={Spectral decomposition of all operators in a {${\rm II}\sb 1$} factor, which is embeddable in ${R}^\omega$},
      NOTE={preprint},
      YEAR={2001},
      }
  • [haageruplarsen] Go to document U. Haagerup and F. Larsen, "Brown’s spectral distribution measure for $R$-diagonal elements in finite von Neumann algebras," J. Funct. Anal., vol. 176, iss. 2, pp. 331-367, 2000.
    @article {haageruplarsen, MRKEY = {1784419},
      AUTHOR = {Haagerup, Uffe and Larsen, Flemming},
      TITLE = {Brown's spectral distribution measure for {$R$}-diagonal elements in finite von {N}eumann algebras},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {176},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {331--367},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {46L54},
      MRNUMBER = {1784419},
      MRREVIEWER = {Dimitri Y. Shlyakhtenko},
      DOI = {10.1006/jfan.2000.3610},
      ZBLNUMBER = {0984.46042},
      }
  • [HS] Go to document U. Haagerup and H. Schultz, "Invariant subspaces for operators in a general ${ II}_1$-factor," Publ. Math. Inst. Hautes Études Sci., iss. 109, pp. 19-111, 2009.
    @article {HS, MRKEY = {2511586},
      AUTHOR = {Haagerup, Uffe and Schultz, Hanne},
      TITLE = {Invariant subspaces for operators in a general {${\rm II}\sb 1$}-factor},
      JOURNAL = {Publ. Math. Inst. Hautes Études Sci.},
      FJOURNAL = {Publications Mathématiques. Institut de Hautes Études Scientifiques},
      NUMBER = {109},
      YEAR = {2009},
      PAGES = {19--111},
      ISSN = {0073-8301},
      MRCLASS = {47A15 (46L10)},
      MRNUMBER = {2511586},
      MRREVIEWER = {{È}. V. Kissin},
      DOI = {10.1007/s10240-009-0018-7},
      ZBLNUMBER = {1178.46058},
      }
  • [HS2] Go to document U. Haagerup and H. Schultz, "Brown measures of unbounded operators affiliated with a finite von Neumann algebra," Math. Scand., vol. 100, iss. 2, pp. 209-263, 2007.
    @article {HS2, MRKEY = {2339369},
      AUTHOR = {Haagerup, Uffe and Schultz, Hanne},
      TITLE = {Brown measures of unbounded operators affiliated with a finite von {N}eumann algebra},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {100},
      YEAR = {2007},
      NUMBER = {2},
      PAGES = {209--263},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {46L54 (46L10 47A15 47B99)},
      MRNUMBER = {2339369},
      MRREVIEWER = {Hari Bercovici},
      ZBLNUMBER = {1168.46039},
      URL={http://www.mscand.dk/article.php?id=3017},
      }
  • [HT] Go to document U. Haagerup and S. Thorbjørnsen, "A new application of random matrices: ${ Ext}(C^*_{ red}(F_2))$ is not a group," Ann. of Math., vol. 162, iss. 2, pp. 711-775, 2005.
    @article {HT, MRKEY = {2183281},
      AUTHOR = {Haagerup, Uffe and Thorbjørnsen, Steen},
      TITLE = {A new application of random matrices: {${\rm Ext}(C\sp *\sb {\rm red}(F\sb 2))$} is not a group},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {162},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {711--775},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {46L54 (15A52 46L35)},
      MRNUMBER = {2183281},
      MRREVIEWER = {Beno\^ ıt Collins},
      DOI = {10.4007/annals.2005.162.711},
      }
  • [horn] Go to document A. Horn, "On the eigenvalues of a matrix with prescribed singular values," Proc. Amer. Math. Soc., vol. 5, pp. 4-7, 1954.
    @article {horn, MRKEY = {0061573},
      AUTHOR = {Horn, Alfred},
      TITLE = {On the eigenvalues of a matrix with prescribed singular values},
      JOURNAL = {Proc. Amer. Math. Soc.},
      FJOURNAL = {Proceedings of the American Mathematical Society},
      VOLUME = {5},
      YEAR = {1954},
      PAGES = {4--7},
      ISSN = {0002-9939},
      MRCLASS = {09.0X},
      MRNUMBER = {0061573},
      MRREVIEWER = {F. Smithies},
      DOI = {10.2307/2032094},
      ZBLNUMBER = {0055.00908},
      }
  • [peresalsbook] B. J. Hough, M. Krishnapur, Y. Peres, and B. Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, Providence, RI: Amer. Math. Soc., 2009, vol. 51.
    @book {peresalsbook, MRKEY = {2552864},
      AUTHOR = {Hough, J. Ben and Krishnapur, Manjunath and Peres, Yuval and Vir{á}g, B{á}lint},
      TITLE = {Zeros of {G}aussian Analytic Functions and Determinantal Point Processes},
      SERIES = {Univ. Lecture Ser.},
      VOLUME = {51},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2009},
      PAGES = {x+154},
      ISBN = {978-0-8218-4373-4},
      MRCLASS = {60G55 (30B20 30C15 60B20 60F10 60G15 65H04 82B31)},
      MRNUMBER = {2552864},
      MRREVIEWER = {Dmitri B. Beliaev},
      ZBLNUMBER = {1190.60038},
      }
  • [sommerslehman] Go to document N. Lehmann and H. Sommers, "Eigenvalue statistics of random real matrices," Phys. Rev. Lett., vol. 67, iss. 8, pp. 941-944, 1991.
    @article {sommerslehman, MRKEY = {1121461},
      AUTHOR = {Lehmann, Nils and Sommers, Hans-J{ü}rgen},
      TITLE = {Eigenvalue statistics of random real matrices},
      JOURNAL = {Phys. Rev. Lett.},
      FJOURNAL = {Physical Review Letters},
      VOLUME = {67},
      YEAR = {1991},
      NUMBER = {8},
      PAGES = {941--944},
      ISSN = {0031-9007},
      CODEN = {PRLTAO},
      MRCLASS = {82B41 (15A18 15A52 82C32)},
      MRNUMBER = {1121461},
      DOI = {10.1103/PhysRevLett.67.941},
      ZBLNUMBER = {0990.82528},
      }
  • [NS] A. Nica and R. Speicher, "$R$-diagonal pairs—a common approach to Haar unitaries and circular elements," in Free Probability Theory, Providence, RI: Amer. Math. Soc., 1997, vol. 12, pp. 149-188.
    @incollection {NS, MRKEY = {1426839},
      AUTHOR = {Nica, Alexandru and Speicher, Roland},
      TITLE = {{$R$}-diagonal pairs---a common approach to {H}aar unitaries and circular elements},
      BOOKTITLE = {Free Probability Theory},
      VENUE={{W}aterloo, {ON},
      1995},
      SERIES = {Fields Inst. Commun.},
      VOLUME = {12},
      PAGES = {149--188},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1997},
      MRCLASS = {46L50},
      MRNUMBER = {1426839},
      MRREVIEWER = {D. Petz},
      ZBLNUMBER = {0889.46053},
      }
  • [panzhou] Go to document G. Pan and W. Zhou, "Circular law, extreme singular values and potential theory," J. Multivariate Anal., vol. 101, iss. 3, pp. 645-656, 2010.
    @article {panzhou, MRKEY = {2575411},
      AUTHOR = {Pan, Guangming and Zhou, Wang},
      TITLE = {Circular law, extreme singular values and potential theory},
      JOURNAL = {J. Multivariate Anal.},
      FJOURNAL = {Journal of Multivariate Analysis},
      VOLUME = {101},
      YEAR = {2010},
      NUMBER = {3},
      PAGES = {645--656},
      ISSN = {0047-259X},
      MRCLASS = {60B20 (15A18 31A10 60F10)},
      MRNUMBER = {2575411},
      DOI = {10.1016/j.jmva.2009.08.005},
      ZBLNUMBER = {1203.60011},
      }
  • [PS] Go to document L. Pastur and M. Shcherbina, "Bulk universality and related properties of Hermitian matrix models," J. Stat. Phys., vol. 130, iss. 2, pp. 205-250, 2008.
    @article {PS, MRKEY = {2375744},
      AUTHOR = {Pastur, L. and Shcherbina, M.},
      TITLE = {Bulk universality and related properties of {H}ermitian matrix models},
      JOURNAL = {J. Stat. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {130},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {205--250},
      ISSN = {0022-4715},
      MRCLASS = {82B41 (82B27)},
      MRNUMBER = {2375744},
      MRREVIEWER = {Florent Benaych-Georges},
      DOI = {10.1007/s10955-007-9434-6},
      ZBLNUMBER = {1136.15015},
      }
  • [SST] Go to document A. Sankar, D. A. Spielman, and S. Teng, "Smoothed analysis of the condition numbers and growth factors of matrices," SIAM J. Matrix Anal. Appl., vol. 28, iss. 2, pp. 446-476, 2006.
    @article {SST, MRKEY = {2255338},
      AUTHOR = {Sankar, Arvind and Spielman, Daniel A. and Teng, Shang-Hua},
      TITLE = {Smoothed analysis of the condition numbers and growth factors of matrices},
      JOURNAL = {SIAM J. Matrix Anal. Appl.},
      FJOURNAL = {SIAM Journal on Matrix Analysis and Applications},
      VOLUME = {28},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {446--476},
      ISSN = {0895-4798},
      MRCLASS = {65F35 (15A12)},
      MRNUMBER = {2255338},
      MRREVIEWER = {Maria C. Adam},
      DOI = {10.1137/S0895479803436202},
      ZBLNUMBER = {1179.65033},
      }
  • [Sniady] Go to document P. Śniady, "Random regularization of Brown spectral measure," J. Funct. Anal., vol. 193, iss. 2, pp. 291-313, 2002.
    @article {Sniady, MRKEY = {1929504},
      AUTHOR = {{\'S}niady, Piotr},
      TITLE = {Random regularization of {B}rown spectral measure},
      JOURNAL = {J. Funct. Anal.},
      FJOURNAL = {Journal of Functional Analysis},
      VOLUME = {193},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {291--313},
      ISSN = {0022-1236},
      CODEN = {JFUAAW},
      MRCLASS = {15A52 (46L53 46L54 60E05)},
      MRNUMBER = {1929504},
      MRREVIEWER = {Wilfried Hazod},
      DOI = {10.1006/jfan.2001.3935},
      ZBLNUMBER = {1026.46056},
      }
  • [taovu] Go to document T. Tao and V. Vu, "Random matrices: the circular law," Commun. Contemp. Math., vol. 10, iss. 2, pp. 261-307, 2008.
    @article {taovu, MRKEY = {2409368},
      AUTHOR = {Tao, Terence and Vu, Van},
      TITLE = {Random matrices: the circular law},
      JOURNAL = {Commun. Contemp. Math.},
      FJOURNAL = {Communications in Contemporary Mathematics},
      VOLUME = {10},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {261--307},
      ISSN = {0219-1997},
      MRCLASS = {60F15 (11P70 15A52 60F05)},
      MRNUMBER = {2409368},
      MRREVIEWER = {Ben Joseph Green},
      DOI = {10.1142/S0219199708002788},
      ZBLNUMBER = {1156.15010},
      }
  • [taovuk] Go to document T. Tao and V. Vu, "Random matrices: universality of ESDs and the circular law," Ann. Probab., vol. 38, iss. 5, pp. 2023-2065, 2010.
    @article {taovuk, MRKEY = {2722794},
      AUTHOR = {Tao, Terence and Vu, Van},
      TITLE = {Random matrices: universality of {ESD}s and the circular law},
      NOTE = {with an appendix by Manjunath Krishnapur},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {38},
      YEAR = {2010},
      NUMBER = {5},
      PAGES = {2023--2065},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {60B20 (60F15 60F17)},
      MRNUMBER = {2722794},
      MRREVIEWER = {Sasha Sodin},
      DOI = {10.1214/10-AOP534},
      ZBLNUMBER = {1203.15025},
      }
  • [Vo91] Go to document D. Voiculescu, "Limit laws for random matrices and free products," Invent. Math., vol. 104, iss. 1, pp. 201-220, 1991.
    @article {Vo91, MRKEY = {1094052},
      AUTHOR = {Voiculescu, Dan},
      TITLE = {Limit laws for random matrices and free products},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {104},
      YEAR = {1991},
      NUMBER = {1},
      PAGES = {201--220},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {46L50 (60F99)},
      MRNUMBER = {1094052},
      MRREVIEWER = {Sh. A. Ayupov},
      DOI = {10.1007/BF01245072},
      ZBLNUMBER = {0736.60007},
      }
  • [VoiculescuAdvances] Go to document D. Voiculescu, "The analogues of entropy and of Fisher’s information measure in free probability theory. VI. Liberation and mutual free information," Adv. Math., vol. 146, iss. 2, pp. 101-166, 1999.
    @article {VoiculescuAdvances, MRKEY = {1711843},
      AUTHOR = {Voiculescu, Dan},
      TITLE = {The analogues of entropy and of {F}isher's information measure in free probability theory. {VI}. {L}iberation and mutual free information},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {146},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {101--166},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {46L54},
      MRNUMBER = {1711843},
      MRREVIEWER = {Dimitri Y. Shlyakhtenko},
      DOI = {10.1006/aima.1998.1819},
      }
  • [weyl] H. Weyl, "Inequalities between the two kinds of eigenvalues of a linear transformation," Proc. Nat. Acad. Sci. U. S. A., vol. 35, pp. 408-411, 1949.
    @article {weyl, MRKEY = {0030693},
      AUTHOR = {Weyl, Hermann},
      TITLE = {Inequalities between the two kinds of eigenvalues of a linear transformation},
      JOURNAL = {Proc. Nat. Acad. Sci. U. S. A.},
      FJOURNAL = {Proceedings of the National Academy of Sciences of the United States of America},
      VOLUME = {35},
      YEAR = {1949},
      PAGES = {408--411},
      ISSN = {0027-8424},
      MRCLASS = {45.0X},
      MRNUMBER = {0030693},
      MRREVIEWER = {F. Smithies},
      ZBLNUMBER = {0032.38701},
      }
  • [zysom] Go to document K. Życzkowski and H. Sommers, "Truncations of random unitary matrices," J. Phys. A, vol. 33, iss. 10, pp. 2045-2057, 2000.
    @article {zysom, MRKEY = {1748745},
      AUTHOR = {{\.Z}yczkowski, Karol and Sommers, Hans-J{ü}rgen},
      TITLE = {Truncations of random unitary matrices},
      JOURNAL = {J. Phys. A},
      FJOURNAL = {Journal of Physics. A. Mathematical and General},
      VOLUME = {33},
      YEAR = {2000},
      NUMBER = {10},
      PAGES = {2045--2057},
      ISSN = {0305-4470},
      CODEN = {JPHAC5},
      MRCLASS = {82B41 (15A52)},
      MRNUMBER = {1748745},
      DOI = {10.1088/0305-4470/33/10/307},
      ZBLNUMBER = {0957.82017},
      }

Authors

Alice Guionnet

UMPA
Ecole Normale Supérieure de Lyon
69264 Lyon
France

Manjunath Krishnapur

Indian Institute of Science
Bangalore
India

Ofer Zeitouni

Department of Mathematics
University of Minnesota
Minneapolis, MN 55455

and

Weizmann Institute of Science,
Rehovot 76100
Israel