Book Review
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MathSciNet review:
2731659
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Book Information:
Authors:
Percy Deift and
Dimitri Gioev
Title:
Random matrix theory: invariant ensembles and universality
Additional book information:
Courant Lecture Notes in Mathematics, 18,
Courant Institute of Mathematical Sciences, New York; American Mathematical Society,
Providence, Rhode Island,
2009,
x+217 pp.,
ISBN 978-0-8218-4737-4,
$33.00
M. J. Ablowitz and H. Segur, Asymptotic solutions of the Korteweg-deVries equation, Studies in Appl. Math. 57 (1976/77), no. 1, 13–44. MR 481656, DOI 10.1002/sapm197757113
Pavel Bleher and Alexander Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324, DOI 10.2307/121101
Peter A. Clarkson and J. Bryce McLeod, A connection formula for the second Painlevé transcendent, Arch. Rational Mech. Anal. 103 (1988), no. 2, 97–138. MR 946971, DOI 10.1007/BF00251504
Percy Deift and Dimitri Gioev, Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices, Comm. Pure Appl. Math. 60 (2007), no. 6, 867–910. MR 2306224, DOI 10.1002/cpa.20164
Percy Deift and Dimitri Gioev, Universality in random matrix theory for orthogonal and symplectic ensembles, Int. Math. Res. Pap. IMRP 2 (2007), Art. ID rpm004, 116. MR 2335245
P. Deift, T. Kriecherbauer, K. T-R McLaughlin, S. Venakides, and X. Zhou, Asymptotics for polynomials orthogonal with respect to varying exponential weights, Internat. Math. Res. Notices 16 (1997), 759–782. MR 1472344, DOI 10.1155/S1073792897000500
P. A. Deift and X. Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math. 48 (1995), no. 3, 277–337. MR 1322812, DOI 10.1002/cpa.3160480304
S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), no. 1, 31–51. MR 555581, DOI 10.1007/BF00283254
Michio Jimbo, Tetsuji Miwa, Yasuko Môri, and Mikio Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Phys. D 1 (1980), no. 1, 80–158. MR 573370, DOI 10.1016/0167-2789(80)90006-8
A. B. J. Kuijlaars and K. T-R McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), no. 6, 736–785. MR 1744002, DOI 10.1002/(SICI)1097-0312(200006)53:6<736::AID-CPA2>3.0.CO;2-5
Madan Lal Mehta, Random matrices, 3rd ed., Pure and Applied Mathematics (Amsterdam), vol. 142, Elsevier/Academic Press, Amsterdam, 2004. MR 2129906
L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), no. 1-2, 109–147. MR 1435193, DOI 10.1007/BF02180200
Alexandre Stojanovic, Universality in orthogonal and symplectic invariant matrix models with quartic potential, Math. Phys. Anal. Geom. 3 (2000), no. 4, 339–373 (2001). MR 1845356, DOI 10.1023/A:1011457714198
Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174. MR 1257246
Craig A. Tracy and Harold Widom, On orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177 (1996), no. 3, 727–754. MR 1385083
Harold Widom, On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94 (1999), no. 3-4, 347–363. MR 1675356, DOI 10.1023/A:1004516918143
References
- M.J. Ablowitz and H. Segur, Asymptotic solutions of the Korteweg-de Vries equation, Stud. Appl. Math.57 (1977) No 1, 13–44. MR 0481656 (58:1757)
- P. Bleher and A. Its, Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model, Ann. of Math. (2) 150 (1999), no. 1, 185–266. MR 1715324 (2000k:42033)
- P. A. Clarkson and J. B. McLeod, A connection formula for the second Painlevé transcendent, Arch. Rational Mech. Anal. 103 (1988), no. 2, 97–138. MR 0946971 (89e:34010)
- P. Deift and D. Gioev, Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices, Comm. Pure Appl. Math. 60 (2007), no. 6, 867–910. MR 2306224 (2008e:60089)
- P. Deift and D. Gioev, Universality in random matrix theory for orthogonal and symplectic ensembles, Int. Math. Res. Pap. IMRP 2007, no. 2, Art. ID. MR 2335245 (2008e:82026)
- P. Deift, T. Kricherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. Pure Appl. Math., 52 (1999), 1335–1425. MR 1472344 (99g:34038)
- P. Deift and X. Zhou, Asymptotics for the Painlevé II equation, Comm. Pure Appl. Math., (48) (1995) no 3, 277–337. MR 1322812 (96d:34004)
- S. P. Hastings and J. B. McLeod, A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rational Mech. Anal. 73 (1980), no. 1, 31–51. MR 0555581 (81i:34024)
- M. Jimbo, T. Miwa, Y. Môri, and M. Sato, Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D 1 (1980), 80-158. MR 0573370 (84k:82037)
- A. B. J. Kuijlaars and K. T.-R. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. Pure Appl. Math. 53 (2000), no. 6, 736–785. MR 1744002 (2001f:31003)
- M. Mehta, Random Matrices, 3rd ed. Elsevier/Academic Press, Amsterdam, 2004. MR 2129906 (2006b:82001)
- L. Pastur and M. Shcherbina, Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86 (1997), no. 1-2, 109–147. MR 1435193 (98b:82037)
- A. Stojanovic, Universality in orthogonal and symplectic invariant matrix models with quartic potential, Math. Phys. Anal. Geom. 3 (2000), no. 4, 339–373; MR 1845356, errata: Math. Phys. Anal. Geom. 7 (2004), no. 4, 347–349. MR 2108624 (2005h:82066)
- C. A. Tracy and H. Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), 33-72. MR 1257246 (95e:82003)
- C. A. Tracy and H. Widom, On orthogonal and symplectic ensembles, Comm. Math. Phys. 177 (1996), 727-754. MR 1385083 (97a:82055)
- H. Widom, On the relation between orthogonal, symplectic and unitary matrix ensembles. J. Statist. Phys. 94 (1999), no. 3-4, 347–363. MR 1675356 (2000e:82024)
Review Information:
Reviewer:
Estelle Basor
Affiliation:
American Institute of Mathematics
Email:
ebasor@aimath.org
Journal:
Bull. Amer. Math. Soc.
48 (2011), 147-152
DOI:
https://doi.org/10.1090/S0273-0979-2010-01307-0
Published electronically:
October 27, 2010
Review copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
