Abstract
We study the asymptotics in $n$ for $n$-dimensional Toeplitz determinants whose symbols possess Fisher-Hartwig singularities on a smooth background. We prove the general nondegenerate asymptotic behavior as conjectured by Basor and Tracy. We also obtain asymptotics of Hankel determinants on a finite interval as well as determinants of Toeplitz+Hankel type. Our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the Riemann-Hilbert approach.
-
[BDJ]
J. Baik, P. Deift, and K. Johansson, "On the distribution of the length of the longest increasing subsequence of random permutations," J. Amer. Math. Soc., vol. 12, iss. 4, pp. 1119-1178, 1999.@article {BDJ, MRKEY = {1682248},
AUTHOR = {Baik, Jinho and Deift, Percy and Johansson, Kurt},
TITLE = {On the distribution of the length of the longest increasing subsequence of random permutations},
JOURNAL = {J. Amer. Math. Soc.},
FJOURNAL = {Journal of the American Mathematical Society},
VOLUME = {12},
YEAR = {1999},
NUMBER = {4},
PAGES = {1119--1178},
ISSN = {0894-0347},
MRCLASS = {05A05 (33D45 45E05 60C05)},
MRNUMBER = {1682248},
MRREVIEWER = {David J. Aldous},
DOI = {10.1090/S0894-0347-99-00307-0},
ZBLNUMBER = {0932.05001},
} -
[BR] J. Baik and E. M. Rains, "Algebraic aspects of increasing subsequences," Duke Math. J., vol. 109, iss. 1, pp. 1-65, 2001.
@article {BR, MRKEY = {1844203},
AUTHOR = {Baik, Jinho and Rains, Eric M.},
TITLE = {Algebraic aspects of increasing subsequences},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {109},
YEAR = {2001},
NUMBER = {1},
PAGES = {1--65},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {05E15 (05A15 05E05 60C05)},
MRNUMBER = {1844203},
MRREVIEWER = {Ira Gessel},
DOI = {10.1215/S0012-7094-01-10911-3},
ZBLNUMBER = {1007.05096},
} -
[Barnes] E. W. Barnes, "The theory of the ${G}$-function," Quart. J. Pure and Appl. Math., vol. 31, pp. 264-313, 1900.
@article{Barnes,
author={Barnes, E. W.},
TITLE={The theory of the ${G}$-function},
JOURNAL={Quart. J. Pure and Appl. Math.},
VOLUME={31},
YEAR={1900},
JFMNUMBER={30.0389.02},
PAGES={264--313},
} -
[B] E. Basor, "Asymptotic formulas for Toeplitz determinants," Trans. Amer. Math. Soc., vol. 239, pp. 33-65, 1978.
@article {B, MRKEY = {0493480},
AUTHOR = {Basor, Estelle},
TITLE = {Asymptotic formulas for {T}oeplitz determinants},
JOURNAL = {Trans. Amer. Math. Soc.},
FJOURNAL = {Transactions of the American Mathematical Society},
VOLUME = {239},
YEAR = {1978},
PAGES = {33--65},
ISSN = {0002-9947},
MRCLASS = {47B35 (42A56)},
MRNUMBER = {0493480},
MRREVIEWER = {J. V. Baxley},
DOI = {10.2307/1997847},
ZBLNUMBER = {0409.47018},
} -
[B2] E. L. Basor, "A localization theorem for Toeplitz determinants," Indiana Univ. Math. J., vol. 28, iss. 6, pp. 975-983, 1979.
@article {B2, MRKEY = {0551161},
AUTHOR = {Basor, E. L.},
TITLE = {A localization theorem for {T}oeplitz determinants},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {28},
YEAR = {1979},
NUMBER = {6},
PAGES = {975--983},
ISSN = {0022-2518},
CODEN = {IUMJAB},
MRCLASS = {47B35},
MRNUMBER = {0551161},
MRREVIEWER = {R. G. Douglas},
DOI = {10.1512/iumj.1979.28.28070},
ZBLNUMBER = {0396.47018},
} -
[BE1] E. L. Basor and T. Ehrhardt, "Some identities for determinants of structured matrices. Special issue on structured and infinite systems of linear equations," Linear Algebra Appl., vol. 343/344, pp. 5-19, 2002.
@article {BE1, MRKEY = {1878935},
AUTHOR = {Basor, Estelle L. and Ehrhardt, Torsten},
TITLE = {Some identities for determinants of structured matrices. Special issue on structured and infinite systems of linear equations},
JOURNAL = {Linear Algebra Appl.},
FJOURNAL = {Linear Algebra and its Applications},
VOLUME = {343/344},
YEAR = {2002},
PAGES = {5--19},
ISSN = {0024-3795},
CODEN = {LAAPAW},
MRCLASS = {15A15 (15A57 47B35)},
MRNUMBER = {1878935},
DOI = {10.1016/S0024-3795(01)00400-1},
ZBLNUMBER = {0994.15008},
} -
[BEsym] E. L. Basor and T. Ehrhardt, "Asymptotic formulas for the determinants of symmetric Toeplitz plus Hankel matrices," in Toeplitz Matrices and Singular Integral Equations, Basel: Birkhäuser, 2002, vol. 135, pp. 61-90.
@incollection {BEsym, MRKEY = {1935758},
AUTHOR = {Basor, Estelle L. and Ehrhardt, Torsten},
TITLE = {Asymptotic formulas for the determinants of symmetric {T}oeplitz plus {H}ankel matrices},
BOOKTITLE = {Toeplitz Matrices and Singular Integral Equations},
VENUE={{P}obershau, 2001},
SERIES = {Oper. Theory Adv. Appl.},
VOLUME = {135},
PAGES = {61--90},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2002},
MRCLASS = {47B35 (15A15 15A57 47N30 82B44)},
MRNUMBER = {1935758},
MRREVIEWER = {A. B{ö}ttcher},
ZBLNUMBER = {1025.15011},
} -
[BEnonsym] E. L. Basor and T. Ehrhardt, "Asymptotic formulas for determinants of a sum of finite Toeplitz and Hankel matrices," Math. Nachr., vol. 228, pp. 5-45, 2001.
@article {BEnonsym, MRKEY = {1845906},
AUTHOR = {Basor, Estelle L. and Ehrhardt, Torsten},
TITLE = {Asymptotic formulas for determinants of a sum of finite {T}oeplitz and {H}ankel matrices},
JOURNAL = {Math. Nachr.},
FJOURNAL = {Mathematische Nachrichten},
VOLUME = {228},
YEAR = {2001},
PAGES = {5--45},
ISSN = {0025-584X},
MRCLASS = {47B35},
MRNUMBER = {1845906},
MRREVIEWER = {N. K. Karapetyants},
DOI = {10.1002/1522-2616(200108)228:1<5::AID-MANA5>3.3.CO;2-5},
ZBLNUMBER = {1047.47020},
} -
[BE4] E. L. Basor and T. Ehrhardt, "Determinant computations for some classes of Toeplitz-Hankel matrices," Oper. Matrices, vol. 3, iss. 2, pp. 167-186, 2009.
@article {BE4, MRKEY = {2522773},
AUTHOR = {Basor, Estelle L. and Ehrhardt, Torsten},
TITLE = {Determinant computations for some classes of {T}oeplitz-{H}ankel matrices},
JOURNAL = {Oper. Matrices},
FJOURNAL = {Operators and Matrices},
VOLUME = {3},
YEAR = {2009},
NUMBER = {2},
PAGES = {167--186},
ISSN = {1846-3886},
MRCLASS = {47B35 (60B20 82B44)},
MRNUMBER = {2522773},
MRREVIEWER = {A. B{ö}ttcher},
ZBLNUMBER = {1198.47042},
} -
[BT] E. L. Basor and C. A. Tracy, "The Fisher-Hartwig conjecture and generalizations," Phys. A, vol. 177, iss. 1-3, pp. 167-173, 1991.
@article {BT, MRKEY = {1137031},
AUTHOR = {Basor, Estelle L. and Tracy, Craig A.},
TITLE = {The {F}isher-{H}artwig conjecture and generalizations},
JOURNAL = {Phys. A},
FJOURNAL = {Physica A. Statistical and Theoretical Physics},
VOLUME = {177},
YEAR = {1991},
NUMBER = {1-3},
PAGES = {167--173},
ISSN = {0378-4371},
CODEN = {PHYSAG},
MRCLASS = {82B05 (47B35)},
MRNUMBER = {1137031},
MRREVIEWER = {Flora Koukiou},
DOI = {10.1016/0378-4371(91)90149-7},
} -
[BE] E. Bateman, Higher Transcendental Functions, New York: McGraw-Hill, 1953.
@book{BE,
author={Bateman, E.},
TITLE={Higher Transcendental Functions},
PUBLISHER={McGraw-Hill},
ADDRESS={New York},
YEAR={1953},
} -
[BS] A. Böttcher and B. Silbermann, "Toeplitz matrices and determinants with Fisher-Hartwig symbols," J. Funct. Anal., vol. 63, iss. 2, pp. 178-214, 1985.
@article {BS, MRKEY = {0803092},
AUTHOR = {B{ö}ttcher, Albrecht and Silbermann, Bernd},
TITLE = {Toeplitz matrices and determinants with {F}isher-{H}artwig symbols},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {63},
YEAR = {1985},
NUMBER = {2},
PAGES = {178--214},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {47B35},
MRNUMBER = {0803092},
MRREVIEWER = {A. Devinatz},
DOI = {10.1016/0022-1236(85)90085-0},
ZBLNUMBER = {0592.47016},
} -
[BS2] A. Böttcher and B. Silbermann, "The asymptotic behavior of Toeplitz determinants for generating functions with zeros of integral orders," Math. Nachr., vol. 102, pp. 79-105, 1981.
@article {BS2, MRKEY = {0642143},
AUTHOR = {B{ö}ttcher, Albrecht and Silbermann, Bernd},
TITLE = {The asymptotic behavior of {T}oeplitz determinants for generating functions with zeros of integral orders},
JOURNAL = {Math. Nachr.},
FJOURNAL = {Mathematische Nachrichten},
VOLUME = {102},
YEAR = {1981},
PAGES = {79--105},
ISSN = {0025-584X},
CODEN = {MTMNAQ},
MRCLASS = {47B35 (15A15 42A38)},
MRNUMBER = {0642143},
MRREVIEWER = {A. Devinatz},
DOI = {10.1002/mana.19811020108},
ZBLNUMBER = {0479.47025},
} -
[BS3] A. Böttcher and B. Silbermann, "Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity," Math. Nachr., vol. 127, pp. 95-123, 1986.
@article {BS3, MRKEY = {0861720},
AUTHOR = {B{ö}ttcher, Albrecht and Silbermann, Bernd},
TITLE = {Toeplitz operators and determinants generated by symbols with one {F}isher-{H}artwig singularity},
JOURNAL = {Math. Nachr.},
FJOURNAL = {Mathematische Nachrichten},
VOLUME = {127},
YEAR = {1986},
PAGES = {95--123},
ISSN = {0025-584X},
CODEN = {MTMNAQ},
MRCLASS = {47B35},
MRNUMBER = {0861720},
MRREVIEWER = {Jeffrey R. Butz},
DOI = {10.1002/mana.19861270108},
ZBLNUMBER = {0613.47024},
} -
[BK] H. M. Bui and J. P. Keating, "On the mean values of $L$-functions in orthogonal and symplectic families," Proc. Lond. Math. Soc., vol. 96, iss. 2, pp. 335-366, 2008.
@article {BK, MRKEY = {2396123},
AUTHOR = {Bui, H. M. and Keating, J. P.},
TITLE = {On the mean values of {$L$}-functions in orthogonal and symplectic families},
JOURNAL = {Proc. Lond. Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {96},
YEAR = {2008},
NUMBER = {2},
PAGES = {335--366},
ISSN = {0024-6115},
MRCLASS = {11M06 (11F66)},
MRNUMBER = {2396123},
MRREVIEWER = {Steven Joel Miller},
DOI = {10.1112/plms/pdm046},
ZBLNUMBER = {05250853},
} -
[DIKadd] P. Deift, A. Its, and I. Krasovsky, On asymptotics of a Toeplitz determinant with singularities.
@misc{DIKadd,
author = {Deift, Percy and Its, A. and Krasovsky, I.},
TITLE={On asymptotics of a {T}oeplitz determinant with singularities},
NOTE={in preparation},
} -
[DZ] P. Deift and X. Zhou, "A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation," Ann. of Math., vol. 137, iss. 2, pp. 295-368, 1993.
@article {DZ, MRKEY = {1207209},
AUTHOR = {Deift, Percy and Zhou, X.},
TITLE = {A steepest descent method for oscillatory {R}iemann-{H}ilbert problems. {A}symptotics for the {MK}d{V} equation},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {137},
YEAR = {1993},
NUMBER = {2},
PAGES = {295--368},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {35Q53 (34A55 34L25 35Q15 35Q55)},
MRNUMBER = {1207209},
MRREVIEWER = {Alexey V. Samokhin},
DOI = {10.2307/2946540},
ZBLNUMBER = {0771.35042},
} -
[Dstrong] P. Deift, T. Kriecherbauer, K. T-R. McLaughlin, S. Venakides, and X. Zhou, "Strong asymptotics of orthogonal polynomials with respect to exponential weights," Comm. Pure Appl. Math., vol. 52, iss. 12, pp. 1491-1552, 1999.
@article {Dstrong, MRKEY = {1711036},
AUTHOR = {Deift, Percy and Kriecherbauer, T. and McLaughlin, K. T-R and Venakides, S. and Zhou, X.},
TITLE = {Strong asymptotics of orthogonal polynomials with respect to exponential weights},
JOURNAL = {Comm. Pure Appl. Math.},
FJOURNAL = {Communications on Pure and Applied Mathematics},
VOLUME = {52},
YEAR = {1999},
NUMBER = {12},
PAGES = {1491--1552},
ISSN = {0010-3640},
CODEN = {CPAMA},
MRCLASS = {42C05},
MRNUMBER = {1711036},
MRREVIEWER = {D. S. Lubinsky},
DOI = {10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.3.CO;2-R},
ZBLNUMBER = {1026.42024},
} -
[D] P. Deift, "Integrable operators," in Differential Operators and Spectral Theory, Providence, RI: Amer. Math. Soc., 1999, vol. 189, pp. 69-84.
@incollection {D, MRKEY = {1730504},
AUTHOR = {Deift, Percy},
TITLE = {Integrable operators},
BOOKTITLE = {Differential Operators and Spectral Theory},
SERIES = {Amer. Math. Soc. Transl. Ser. 2},
VOLUME = {189},
PAGES = {69--84},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1999},
MRCLASS = {47G10 (35Q15 45P05 47B35 82-02)},
MRNUMBER = {1730504},
MRREVIEWER = {Luen-Chau Li},
ZBLNUMBER = {0920.35105},
} -
[ES] T. Ehrhardt and B. Silbermann, "Toeplitz determinants with one Fisher-Hartwig singularity," J. Funct. Anal., vol. 148, iss. 1, pp. 229-256, 1997.
@article {ES, MRKEY = {1461501},
AUTHOR = {Ehrhardt, Torsten and Silbermann, Bernd},
TITLE = {Toeplitz determinants with one {F}isher-{H}artwig singularity},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {148},
YEAR = {1997},
NUMBER = {1},
PAGES = {229--256},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {47B35 (15A15)},
MRNUMBER = {1461501},
MRREVIEWER = {Estelle L. Basor},
DOI = {10.1006/jfan.1996.3074},
ZBLNUMBER = {0909.47019},
} -
[Ehr] T. Ehrhardt, "A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities," in Recent Advances in Operator Theory, Basel: Birkhäuser, 2001, vol. 124, pp. 217-241.
@incollection {Ehr, MRKEY = {1839838},
AUTHOR = {Ehrhardt, Torsten},
TITLE = {A status report on the asymptotic behavior of {T}oeplitz determinants with {F}isher-{H}artwig singularities},
BOOKTITLE = {Recent Advances in Operator Theory},
VENUE={{G}roningen, 1998},
SERIES = {Oper. Theory Adv. Appl.},
VOLUME = {124},
PAGES = {217--241},
PUBLISHER = {Birkhäuser},
ADDRESS = {Basel},
YEAR = {2001},
MRCLASS = {47B35 (15A57)},
MRNUMBER = {1839838},
MRREVIEWER = {Shao Kuan Li},
ZBLNUMBER = {0993.47028},
} -
[FA] F. Franchini and A. G. Abanov, "Asymptotics of Toeplitz determinants and the emptiness formation probability for the $XY$ spin chain," J. Phys. A, vol. 38, iss. 23, pp. 5069-5095, 2005.
@article {FA, MRKEY = {2148411},
AUTHOR = {Franchini, Fabio and Abanov, Alexander G.},
TITLE = {Asymptotics of {T}oeplitz determinants and the emptiness formation probability for the {$XY$} spin chain},
JOURNAL = {J. Phys. A},
FJOURNAL = {Journal of Physics. A. Mathematical and General},
VOLUME = {38},
YEAR = {2005},
NUMBER = {23},
PAGES = {5069--5095},
ISSN = {0305-4470},
CODEN = {JPHAC5},
MRCLASS = {82B20 (47B35 47N55 82B27)},
MRNUMBER = {2148411},
DOI = {10.1088/0305-4470/38/23/002},
ZBLNUMBER = {02190658},
} -
[FH] M. E. Fischer and R. E. Hartwig, "Toeplitz determinants: Some applications, theorems, and conjectures," Advan. Chem. Phys., vol. 15, pp. 333-353, 1968.
@article{FH,
author={Fischer, M. E. and Hartwig, R. E.},
TITLE={Toeplitz determinants: {S}ome applications, theorems, and conjectures},
JOURNAL={Advan. Chem. Phys.},
VOLUME={15},
YEAR={1968},
PAGES={333--353},
} -
[FIK] A. S. Fokas, A. R. Its, and A. V. Kitaev, "The isomonodromy approach to matrix models in $2$D quantum gravity," Comm. Math. Phys., vol. 147, iss. 2, pp. 395-430, 1992.
@article {FIK, MRKEY = {1174420},
AUTHOR = {Fokas, A. S. and Its, A. R. and Kitaev, A. V.},
TITLE = {The isomonodromy approach to matrix models in {$2$}{D} quantum gravity},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {147},
YEAR = {1992},
NUMBER = {2},
PAGES = {395--430},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {81T40 (58F07)},
MRNUMBER = {1174420},
MRREVIEWER = {Yolanda Lozano},
DOI={10.1007/BF02096594},
ZBLNUMBER={0760.35051},
} -
[FF] P. J. Forrester and N. E. Frankel, "Applications and generalizations of Fisher-Hartwig asymptotics," J. Math. Phys., vol. 45, iss. 5, pp. 2003-2028, 2004.
@article {FF, MRKEY = {2054146},
AUTHOR = {Forrester, P. J. and Frankel, N. E.},
TITLE = {Applications and generalizations of {F}isher-{H}artwig asymptotics},
JOURNAL = {J. Math. Phys.},
FJOURNAL = {Journal of Mathematical Physics},
VOLUME = {45},
YEAR = {2004},
NUMBER = {5},
PAGES = {2003--2028},
ISSN = {0022-2488},
CODEN = {JMAPAQ},
MRCLASS = {82B20 (15A15 15A52 47B35)},
MRNUMBER = {2054146},
MRREVIEWER = {Alexei Khorunzhy},
DOI = {10.1063/1.1699484},
ZBLNUMBER = {1071.82014},
} -
[Geronimo] J. S. Geronimo, "Szegő’s theorem for Hankel determinants," J. Math. Phys., vol. 20, iss. 3, pp. 484-491, 1979.
@article {Geronimo, MRKEY = {0525110},
AUTHOR = {Geronimo, J. S.},
TITLE = {Szegő's theorem for {H}ankel determinants},
JOURNAL = {J. Math. Phys.},
FJOURNAL = {Journal of Mathematical Physics},
VOLUME = {20},
YEAR = {1979},
NUMBER = {3},
PAGES = {484--491},
ISSN = {0022-2488},
CODEN = {JMAPAQ},
MRCLASS = {42A70 (41A60)},
MRNUMBER = {0525110},
MRREVIEWER = {W. A. Al-Salam},
DOI = {10.1063/1.524099},
} -
[GHK] S. M. Gonek, C. P. Hughes, and J. P. Keating, "A hybrid Euler-Hadamard product for the Riemann zeta function," Duke Math. J., vol. 136, iss. 3, pp. 507-549, 2007.
@article {GHK, MRKEY = {2309173},
AUTHOR = {Gonek, S. M. and Hughes, C. P. and Keating, J. P.},
TITLE = {A hybrid {E}uler-{H}adamard product for the {R}iemann zeta function},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {136},
YEAR = {2007},
NUMBER = {3},
PAGES = {507--549},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {11M26 (11M06 15A52)},
MRNUMBER = {2309173},
MRREVIEWER = {Steven Joel Miller},
URL = {http://projecteuclid.org/getRecord?id=euclid.dmj/1170084897},
ZBLNUMBER = {1171.11049},
} -
[IK] A. Its and I. Krasovsky, "Hankel determinant and orthogonal polynomials for the Gaussian weight with a jump," in Integrable Systems and Random Matrices, Providence, RI: Amer. Math. Soc., 2008, vol. 458, pp. 215-247.
@incollection {IK, MRKEY = {2411909},
AUTHOR = {Its, A. and Krasovsky, I.},
TITLE = {Hankel determinant and orthogonal polynomials for the {G}aussian weight with a jump},
BOOKTITLE = {Integrable Systems and Random Matrices},
SERIES = {Contemp. Math.},
VOLUME = {458},
PAGES = {215--247},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2008},
MRCLASS = {47B35 (30E25 34E05 34M50 42C05)},
MRNUMBER = {2411909},
MRREVIEWER = {A. B{ö}ttcher},
ZBLNUMBER = {1163.15027},
} -
[J2] K. Johansson, "On random matrices from the compact classical groups," Ann. of Math., vol. 145, iss. 3, pp. 519-545, 1997.
@article {J2, MRKEY = {1454702},
AUTHOR = {Johansson, Kurt},
TITLE = {On random matrices from the compact classical groups},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {145},
YEAR = {1997},
NUMBER = {3},
PAGES = {519--545},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {60B15 (15A52 47B35 60F05 82B05)},
MRNUMBER = {1454702},
MRREVIEWER = {A. B{ö}ttcher},
DOI = {10.2307/2951843},
ZBLNUMBER = {0883.60010},
} -
[Keating1] J. P. Keating, "Applications of random matrices in physics," in Proceedings of the NATO Adv. Study Inst., New York: Springer-Verlag, 2006, vol. 221, p. x.
@incollection {Keating1, MRKEY = {2238825},
AUTHOR={Keating, J. P.},
TITLE = {Applications of random matrices in physics},
BOOKTITLE = {Proceedings of the {NATO} {A}dv. {S}tudy {I}nst.},
SERIES = {NATO Science Series II: Mathematics, Physics and Chemistry},
VOLUME = {221},
NOTE = {edited by {É}douard Br{é}zin, Vladimir Kazakov, Didina Serban, Paul Wiegmann and Anton Zabrodin},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2006},
PAGES = {x+513},
ISBN = {978-1-4020-4529-5; 1-4020-4529-8},
MRCLASS = {82-06 (15-06 60-06 81-06)},
MRNUMBER = {2238825},
DOI = {10.1007/1-4020-4531-X},
} -
[KM] J. P. Keating and F. Mezzadri, "Random matrix theory and entanglement in quantum spin chains," Comm. Math. Phys., vol. 252, iss. 1-3, pp. 543-579, 2004.
@article {KM, MRKEY = {2104889},
AUTHOR = {Keating, J. P. and Mezzadri, F.},
TITLE = {Random matrix theory and entanglement in quantum spin chains},
JOURNAL = {Comm. Math. Phys.},
FJOURNAL = {Communications in Mathematical Physics},
VOLUME = {252},
YEAR = {2004},
NUMBER = {1-3},
PAGES = {543--579},
ISSN = {0010-3616},
CODEN = {CMPHAY},
MRCLASS = {82B44 (33E17 47B35 47N55 82B41)},
MRNUMBER = {2104889},
DOI = {10.1007/s00220-004-1188-2},
ZBLNUMBER = {1124.82009},
} -
[Kduke] I. V. Krasovsky, "Correlations of the characteristic polynomials in the Gaussian unitary ensemble or a singular Hankel determinant," Duke Math. J., vol. 139, iss. 3, pp. 581-619, 2007.
@article {Kduke, MRKEY = {2350854},
AUTHOR = {Krasovsky, I. V.},
TITLE = {Correlations of the characteristic polynomials in the {G}aussian unitary ensemble or a singular {H}ankel determinant},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {139},
YEAR = {2007},
NUMBER = {3},
PAGES = {581--619},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {15A52 (41A60 47B35 82B44)},
MRNUMBER = {2350854},
DOI = {10.1215/S0012-7094-07-13936-X},
ZBLNUMBER = {1173.15012},
} -
[KVA] A. B. J. Kuijlaars, K. T. -R. McLaughlin, W. Van Assche, and M. Vanlessen, "The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on $[-1,1]$," Adv. Math., vol. 188, iss. 2, pp. 337-398, 2004.
@article {KVA, MRKEY = {2087231},
AUTHOR = {Kuijlaars, A. B. J. and McLaughlin, K. T.-R. and Van Assche, W. and Vanlessen, M.},
TITLE = {The {R}iemann-{H}ilbert approach to strong asymptotics for orthogonal polynomials on {$[-1,1]$}},
JOURNAL = {Adv. Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {188},
YEAR = {2004},
NUMBER = {2},
PAGES = {337--398},
ISSN = {0001-8708},
CODEN = {ADMTA4},
MRCLASS = {33C45 (30E25 37K15 42C05)},
MRNUMBER = {2087231},
MRREVIEWER = {Heinrich Begehr},
DOI = {10.1016/j.aim.2003.08.015},
ZBLNUMBER = {1082.42017},
} -
[L] A. Lenard, "Momentum distribution in the ground state of the one-dimensional system of impenetrable Bosons," J. Mathematical Phys., vol. 5, pp. 930-943, 1964.
@article {L, MRKEY = {0171550},
AUTHOR = {Lenard, A.},
TITLE = {Momentum distribution in the ground state of the one-dimensional system of impenetrable {B}osons},
JOURNAL = {J. Mathematical Phys.},
FJOURNAL = {Journal of Mathematical Physics},
VOLUME = {5},
YEAR = {1964},
PAGES = {930--943},
ISSN = {0022-2488},
MRCLASS = {81.34},
MRNUMBER = {0171550},
MRREVIEWER = {D. J. Thouless},
DOI = {10.1063/1.1704196},
} -
[MMS1] A. Mart’inez-Finkelshtein, K. T. -R. McLaughlin, and E. B. Saff, "Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics," Constr. Approx., vol. 24, iss. 3, pp. 319-363, 2006.
@article {MMS1, MRKEY = {2253965},
AUTHOR = {Mart{\'ı}nez-Finkelshtein, A. and McLaughlin, K. T.-R. and Saff, E. B.},
TITLE = {Szegő orthogonal polynomials with respect to an analytic weight: canonical representation and strong asymptotics},
JOURNAL = {Constr. Approx.},
FJOURNAL = {Constructive Approximation. An International Journal for Approximations and Expansions},
VOLUME = {24},
YEAR = {2006},
NUMBER = {3},
PAGES = {319--363},
ISSN = {0176-4276},
MRCLASS = {42C05 (30E10 30E15 41A35 41A60)},
MRNUMBER = {2253965},
MRREVIEWER = {R. A. Zalik},
DOI = {10.1007/s00365-005-0617-6},
} -
[MMS2] A. Mart’inez-Finkelshtein, K. T. -R. McLaughlin, and E. B. Saff, "Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle," Int. Math. Res. Not., vol. 2006, p. I, 2006.
@article {MMS2, MRKEY = {2250012},
AUTHOR = {Mart{\'ı}nez-Finkelshtein, A. and McLaughlin, K. T.-R. and Saff, E. B.},
TITLE = {Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle},
JOURNAL = {Int. Math. Res. Not.},
FJOURNAL = {International Mathematics Research Notices},
YEAR = {2006},
PAGES = {Art. ID 91426, 43},
ISSN = {1073-7928},
MRCLASS = {42C05 (30C10 33C10 41A60 47B35)},
MRNUMBER = {2250012},
MRREVIEWER = {Juan Jos{é} Moreno-Balc{á}zar},
DOI = {10.1155/IMRN/2006/91426},
VOLUME = {2006},
} -
[Ovchinnikov] A. A. Ovchinnikov, "Fisher-Hartwig conjecture and the correlators in the impenetrable Bose gas," Phys. Lett. A, vol. 373, iss. 3, pp. 305-307, 2009.
@article {Ovchinnikov, MRKEY = {2488492},
AUTHOR = {Ovchinnikov, A. A.},
TITLE = {Fisher-{H}artwig conjecture and the correlators in the impenetrable {B}ose gas},
JOURNAL = {Phys. Lett. A},
FJOURNAL = {Physics Letters. A},
VOLUME = {373},
YEAR = {2009},
NUMBER = {3},
PAGES = {305--307},
ISSN = {0375-9601},
CODEN = {PYLAAG},
MRCLASS = {82B10},
MRNUMBER = {2488492},
DOI = {10.1016/j.physleta.2008.11.032},
} -
[Simon] B. Simon, Orthogonal Polynomials on the Unit Circle, Providence, RI: Amer. Math. Soc., 2005.
@book{Simon,
author={Simon, B.},
TITLE={Orthogonal Polynomials on the Unit Circle},
SERIES={Amer. Math. Soc. Colloq. Publ.},
PUBLISHER={Amer. Math. Soc.},
ADDRESS={Providence, RI},
YEAR={2005},
} -
[Szego] G. Szegö, Orthogonal Polynomials, Providence, R.I.: Amer. Math. Soc., 1959, vol. 23.
@book {Szego, MRKEY = {0106295},
AUTHOR = {Szeg{ö},
Gabor},
TITLE = {Orthogonal Polynomials},
SERIES = {Amer. Math. Soc. Colloq. Publ.},
VOLUME={23},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, R.I.},
YEAR = {1959},
PAGES = {ix+421},
MRCLASS = {33.00 (41.00)},
MRNUMBER = {0106295},
MRREVIEWER = {B. Germansky},
ZBLNUMBER = {0089.27501},
} -
[Weyl] H. Weyl, The Classical Groups, Princeton, NJ: Princeton Univ. Press, 1997.
@book {Weyl, MRKEY = {1488158},
AUTHOR = {Weyl, Hermann},
TITLE = {The Classical Groups},
SERIES = {Princeton Landmarks in Math.},
PUBLISHER = {Princeton Univ. Press},
ADDRESS = {Princeton, NJ},
YEAR = {1997},
PAGES = {xiv+320},
ISBN = {0-691-05756-7},
MRCLASS = {01A75 (20-03 20Cxx 20Gxx 22-01)},
MRNUMBER = {1488158},
ZBLNUMBER = {1024.20501},
} -
[WW] E. Whittaker and G. Watson, A Course of Modern Analysis, Cambridge: Cambridge University Press, 1996.
@book{WW,
author={Whittaker, E. and Watson, G.},
TITLE={A Course of Modern Analysis},
SERIES = {Cambridge Math. Library},
PUBLISHER = {Cambridge University Press},
ADDRESS = {Cambridge},
YEAR = {1996},
PAGES = {vi+608},
ISBN = {0-521-58807-3},
MRCLASS = {01A75 (30-01 33-01)},
MRNUMBER = {1424469},
} -
[W] H. Widom, "Toeplitz determinants with singular generating functions," Amer. J. Math., vol. 95, pp. 333-383, 1973.
@article {W, MRKEY = {0331107},
AUTHOR = {Widom, Harold},
TITLE = {Toeplitz determinants with singular generating functions},
JOURNAL = {Amer. J. Math.},
FJOURNAL = {American Journal of Mathematics},
VOLUME = {95},
YEAR = {1973},
PAGES = {333--383},
ISSN = {0002-9327},
MRCLASS = {47B35 (45E10)},
MRNUMBER = {0331107},
MRREVIEWER = {I. I. Hirschman, Jr.},
DOI = {10.2307/2373789},
ZBLNUMBER = {0275.45006},
} -
[Warc] H. Widom, "The strong Szegő limit theorem for circular arcs," Indiana Univ. Math. J., vol. 21, pp. 277-283, 1971/1972.
@article {Warc, MRKEY = {0288495},
AUTHOR = {Widom, Harold},
TITLE = {The strong {S}zegő limit theorem for circular arcs},
JOURNAL = {Indiana Univ. Math. J.},
FJOURNAL = {Indiana University Mathematics Journal},
VOLUME = {21},
YEAR = {1971/1972},
PAGES = {277--283},
ISSN = {0022-2518},
MRCLASS = {42.10},
MRNUMBER = {0288495},
MRREVIEWER = {I. I. Hirschman, Jr.},
DOI = {10.1512/iumj.1971.21.21022},
}
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