메뉴 건너뛰기




Volumn 12, Issue 4, 1999, Pages 1119-1178

On the distribution of the length of the longest increasing subsequence of random permutations

Author keywords

Orthogonal polynomials; Random matrices; Random permutations; Riemann Hilbert problems; Steepest descent method

Indexed keywords


EID: 0033443150     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0894-0347-99-00307-0     Document Type: Article
Times cited : (980)

References (62)
  • 2
    • 51249162954 scopus 로고
    • Hammersley's interacting particle process and longest increasing subsequences
    • MR 96k:60017
    • D.Aldous and P.Diaconis, Hammersley's Interacting Particle Process and Longest Increasing Subsequences, Prob. Th. and Rel. Fields, 103, 199-213, (1995). MR 96k:60017
    • (1995) Prob. Th. and Rel. Fields , vol.103 , pp. 199-213
    • Aldous, D.1    Diaconis, P.2
  • 3
    • 84968520048 scopus 로고
    • Natural sorting over permutation spaces
    • MR 37:3800
    • R.M.Baer and P.Brock, Natural sorting over permutation spaces, Math. Comp., 22, 385-410, (1968). MR 37:3800
    • (1968) Math. Comp. , vol.22 , pp. 385-410
    • Baer, R.M.1    Brock, P.2
  • 6
    • 84990617857 scopus 로고
    • Scattering and inverse scattering for first order systems
    • MR 85f:34020
    • R.Beals and R.Coifman, Scattering and inverse scattering for first order systems, Comm. Pure Appl. Math., 37, 39-90, (1984). MR 85f:34020
    • (1984) Comm. Pure Appl. Math. , vol.37 , pp. 39-90
    • Beals, R.1    Coifman, R.2
  • 7
    • 0009270029 scopus 로고    scopus 로고
    • Longest increasing subsequences of random colored permutations, electron
    • CMP 99:07
    • A.Borodin, Longest increasing subsequences of random colored permutations, Electron. J. Combin., 6 (1), R13, (1999). CMP 99:07
    • (1999) J. Combin. , vol.6 , Issue.1
    • Borodin, A.1
  • 9
    • 17744362855 scopus 로고    scopus 로고
    • Integrable hamiltonian systems. Dynamical systems and probabilistic methods in partial differential equations, 103-138
    • edited by P.A.Deift, C.D.Levermore and C.E.Wayne, American Mathematical Society, Providence, RI, MR 96i:58071
    • P.A.Deift, Integrable Hamiltonian systems. Dynamical systems and probabilistic methods in partial differential equations, 103-138, in Lectures in Applied Mathematics, 31, edited by P.A.Deift, C.D.Levermore and C.E.Wayne, American Mathematical Society, Providence, RI, 1996. MR 96i:58071
    • (1996) Lectures in Applied Mathematics , vol.31
    • Deift, P.A.1
  • 11
    • 0000743268 scopus 로고    scopus 로고
    • New results on the equilibrium measure for logarithmic potentials in the presence of an external field
    • CMP 99:05
    • P.A.Deift, T.Kriecherbauer and K.T-R McLaughlin, New Results on the Equilibrium Measure for Logarithmic potentials in the Presence of an External Field, J. Approx. Theory, 95, no.3, 388-475, (1998). CMP 99:05
    • (1998) J. Approx. Theory , vol.95 , Issue.3 , pp. 388-475
    • Deift, P.A.1    Kriecherbauer, T.2    McLaughlin, K.T.-R.3
  • 12
    • 1842689760 scopus 로고    scopus 로고
    • Asymptotics for polynomials orthogonal with respect to varying exponential weights, internat
    • MR 99g:34038
    • P.A.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, no. 16, 759-782, (1997). MR 99g:34038
    • (1997) Math. Res. Notices , Issue.16 , pp. 759-782
    • Deift, P.A.1    Kriecherbauer, T.2    McLaughlin, K.T.-R.3    Venakides, S.4    Zhou, X.5
  • 13
    • 0038887882 scopus 로고    scopus 로고
    • Strong asymptotics for orthogonal polynomials with respect to varying exponential weights via riemann-hilbert techniques
    • P.A.Deift, T.Kriecherbauer, K.T-R McLaughlin, S.Venakides and X.Zhou, Strong Asymptotics for Orthogonal Polynomials with respect to Varying Exponential Weights via Riemann-Hilbert Techniques, To appear in Comm. Pure. Appl. Math.
    • Comm. Pure. Appl. Math.
    • Deift, P.A.1    Kriecherbauer, T.2    McLaughlin, K.T.-R.3    Venakides, S.4    Zhou, X.5
  • 14
    • 0033440723 scopus 로고    scopus 로고
    • Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
    • P.A.Deift, T.Kriecherbauer, 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, To appear in Comm. Pure. Appl. Math.
    • Comm. Pure. Appl. Math.
    • Deift, P.A.1    Kriecherbauer, T.2    McLaughlin, K.T.-R.3    Venakides, S.4    Zhou, X.5
  • 15
    • 84990578228 scopus 로고
    • The collisionless shock region for the long-time behavior of solutions of the KdV equation
    • MR 95f:35220
    • P.A.Deift, S.Venakides and X.Zhou, The collisionless shock region for the long-time behavior of solutions of the KdV equation, Comm. Pure Appl. Math., 47 no. 2, 199-206, (1994). MR 95f:35220
    • (1994) Comm. Pure Appl. Math. , vol.47 , Issue.2 , pp. 199-206
    • Deift, P.A.1    Venakides, S.2    Zhou, X.3
  • 16
    • 1842665167 scopus 로고    scopus 로고
    • New results in small dispersion KdV by an extension of the steepest descent method for Riernann-Hilbert problems
    • MR 98b:35155
    • P.A.Deift, S.Venakides and X.Zhou, New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riernann-Hilbert Problems, Internat. Math. Res. Notices, no. 6, 285-299, (1997). MR 98b:35155
    • (1997) Internat. Math. Res. Notices , Issue.6 , pp. 285-299
    • Deift, P.A.1    Venakides, S.2    Zhou, X.3
  • 17
    • 0001525213 scopus 로고
    • A steepest descent method for oscillatory riemman-hilbert problems; asymptotics for the MKdV equation
    • MR 94d:35143
    • P.A.Deift and X.Zhou, A Steepest Descent Method for Oscillatory Riemman-Hilbert Problems; Asymptotics for the MKdV Equation, Ann. Math., 137, 295-368, (1993). MR 94d:35143
    • (1993) Ann. Math. , vol.137 , pp. 295-368
    • Deift, P.A.1    Zhou, X.2
  • 18
    • 84990647031 scopus 로고
    • Asymptotics for the painlevé II equation
    • MR 96d:34004
    • P.A.Deift and X.Zhou, Asymptotics for the Painlevé II Equation, Comm. Pure. Appl. Math., 48, 277-337, (1995). MR 96d:34004
    • (1995) Comm. Pure. Appl. Math. , vol.48 , pp. 277-337
    • Deift, P.A.1    Zhou, X.2
  • 19
    • 0001368538 scopus 로고
    • Limiting curves for i.I.D. Records
    • MR 96h:60086
    • J.-D.Deuschel and O.Zeitouni, Limiting curves for i.i.d. records, Ann. Probab., 23, 852-878, (1995). MR 96h:60086
    • (1995) Ann. Probab. , vol.23 , pp. 852-878
    • Deuschel, J.-D.1    Zeitouni, O.2
  • 21
    • 0001891773 scopus 로고
    • On the eigenvalues of random matrices
    • MR 95m:60011
    • P.Diaconis and M.Shahshahani, On the Eigenvalues of Random matrices, J. Appl. Prob. 31, 49-61, (1994). MR 95m:60011
    • (1994) J. Appl. Prob. , vol.31 , pp. 49-61
    • Diaconis, P.1    Shahshahani, M.2
  • 22
    • 0001086798 scopus 로고
    • A combinatorial theorem in geometry
    • P.Erdös and G.Szekeres, A combinatorial theorem in geometry, Compositio Math., 2, 463-470, (1935).
    • (1935) Compositio Math. , vol.2 , pp. 463-470
  • 23
    • 34250256469 scopus 로고
    • Monodromy and spectrum preserving deformations, I
    • MR 82g:35103
    • H.Flaschka and A.Newell, Monodromy and spectrum preserving deformations, I, Comm. Math. Phy., 76, no.1, 67-116, (1980). MR 82g:35103
    • (1980) Comm. Math. Phy. , vol.76 , Issue.1 , pp. 67-116
    • Flaschka, H.1    Newell, A.2
  • 24
    • 0000533283 scopus 로고
    • Discrete painlevé equations and their appearance in quantum gravity
    • MR 93a:58080
    • A.S.Fokas, A.R.Its and V.E.Kitaev, Discrete Painlevé equations and their appearance in quantum gravity, Comm. Math. Phy., 142, 313-344, (1991). MR 93a:58080
    • (1991) Comm. Math. Phy. , vol.142 , pp. 313-344
    • Fokas, A.S.1    Its, A.R.2    Kitaev, V.E.3
  • 25
    • 0001348241 scopus 로고
    • On the solvability of painlevé I, III and V
    • MR 93h:35154
    • A.S.Fokas, U.Mugan and X.Zhou, On the Solvability of Painlevé I, III and V, Inverse Problems, 8, 757-785, (1992). MR 93h:35154
    • (1992) Inverse Problems , vol.8 , pp. 757-785
    • Fokas, A.S.1    Mugan, U.2    Zhou, X.3
  • 26
    • 0000528846 scopus 로고
    • On the solvability of painlevé II and IV
    • MR 93d:34004
    • A.S.Fokas and X.Zhou, On the Solvability of Painlevé II and IV, Comm. Math. Phy., 144, 601-622, (1992). MR 93d:34004
    • (1992) Comm. Math. Phy. , vol.144 , pp. 601-622
    • Fokas, A.S.1    Zhou, X.2
  • 27
    • 0003166713 scopus 로고
    • Symmetric functions and P-recursiveness
    • MR 91c:05190
    • I.M.Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory. Ser. A, 53, 257-285, (1990). MR 91c:05190
    • (1990) J. Combin. Theory. Ser. A , vol.53 , pp. 257-285
    • Gessel, I.M.1
  • 29
    • 0001787062 scopus 로고
    • Operator theory, advances and applications; v. Birkhäuser Verlag, Basel, MR 93c:47061; MR 93k:47059
    • I.Gohberg and N.Krupnik, One-Dimensional Linear Singular Integral Equations vol. I and II, Operator theory, advances and applications; v. 53-54, Birkhäuser Verlag, Basel, 1992 MR 93c:47061; MR 93k:47059
    • (1992) One-Dimensional Linear Singular Integral Equations , vol.1-2 , pp. 53-54
    • Gohberg, I.1    Krupnik, N.2
  • 30
    • 33744544553 scopus 로고
    • Possible third-order phase transition in the large N lattice gauge theory
    • D.J.Gross and E.Witten, Possible third-order phase transition in the large N lattice gauge theory, Phys. Rev. D, 21, 446-453, (1980).
    • (1980) Phys. Rev. D , vol.21 , pp. 446-453
    • Gross, D.J.1    Witten, E.2
  • 32
    • 0019147547 scopus 로고
    • A boundary value problem associated with the second painlevé transcendent and the Korteweg de Vries equation
    • MR 81i:34024
    • 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, 31-51, (1980). MR 81i:34024
    • (1980) Arch. Rational Mech. Anal. , vol.73 , pp. 31-51
    • Hastings, S.P.1    McLeod, J.B.2
  • 33
    • 0001292292 scopus 로고    scopus 로고
    • Unitary matrix models and painlevé III
    • MR 97h:81200
    • M.Hisakado, Unitary matrix models and Painlevé III, Modern Phys. Lett. A, 11, no.38, 3001-3010, (1996). MR 97h:81200
    • (1996) Modern Phys. Lett. A , vol.11 , Issue.38 , pp. 3001-3010
    • Hisakado, M.1
  • 34
    • 0003219756 scopus 로고
    • The isomonodromic deformation method in the theory of painlevé equations
    • Springer-Verlag, Berlin, Heidelberg, MR 89b:34012
    • A.R.Its and V.Yu.Novokshenov, The Isomonodromic Deformation Method in the Theory of Painlevé Equations, Lecture Notes in Math.1191, Springer-Verlag, Berlin, Heidelberg, 1986. MR 89b:34012
    • (1986) Lecture Notes in Math. , vol.1191
    • Its, A.R.1    Novokshenov, V.Y.2
  • 35
    • 49149137872 scopus 로고
    • Monodromy preserving deformations of linear ordinary differential equations with rational coefficients, I. General theory and τ-function
    • MR 83k:34010a
    • M.Jimbo, T.Miwa and K.Ueno, Monodromy preserving deformations of linear ordinary differential equations with rational coefficients, I. General theory and τ-function, Physica D, 2, no.2, 306-352, (1981). MR 83k:34010a
    • (1981) Physica D , vol.2 , Issue.2 , pp. 306-352
    • Jimbo, M.1    Miwa, T.2    Ueno, K.3
  • 36
    • 0032219317 scopus 로고    scopus 로고
    • The longest increasing subsequence in a random permutation and a unitary random matrix model
    • MR 99e:60033
    • K.Johansson, The Longest Increasing Subsequence in a Random Permutation and a Unitary Random Matrix Model, Math. Res. Lett., 5, no.1-2, 63-82, (1998). MR 99e:60033
    • (1998) Math. Res. Lett. , vol.5 , Issue.1-2 , pp. 63-82
    • Johansson, K.1
  • 39
    • 21144469473 scopus 로고
    • On the long time behavior of the doubly infinite toda lattice under initial data decaying at infinity
    • MR 94c:58086
    • S.Kamvissis, On the Long Time Behavior of the Doubly Infinite Toda Lattice under Initial Data Decaying at Infinity, Comm. Math. Phy., 153, 479-519, (1993). MR 94c:58086
    • (1993) Comm. Math. Phy. , vol.153 , pp. 479-519
    • Kamvissis, S.1
  • 40
    • 0001163892 scopus 로고    scopus 로고
    • On the longest increasing subsequence of random permutations - A concentration result
    • J.-H. Kim, On the longest increasing subsequence of random permutations - a concentration result, J. Comb. Th. A, vol. 76, 148-155, (1996).
    • (1996) J. Comb. Th. A , vol.76 , pp. 148-155
    • Kim, J.-H.1
  • 41
    • 0003657590 scopus 로고
    • sorting and searching, 2nd ed., Addison Wesley, Reading, Mass., MR 56:4281
    • D.E.Knuth, The art of computer programming, vol. 3: sorting and searching, 2nd ed., Addison Wesley, Reading, Mass., 1973. MR 56:4281
    • (1973) The Art of Computer Programming , vol.3
    • Knuth, D.E.1
  • 42
    • 0001345688 scopus 로고
    • A variational problem for random Young tableaux
    • MR 98e:05108
    • B.F.Logan and L.A.Shepp, A variational problem for random Young tableaux, Advances in Math., 26, 206-222, (1977). MR 98e:05108
    • (1977) Advances in Math. , vol.26 , pp. 206-222
  • 47
    • 0347366076 scopus 로고
    • Unitary-Matrix models as exactly solvable string theories
    • V.Periwal and D.Shevitz, Unitary-Matrix Models as Exactly Solvable String Theories, Phys. Rev. Lett., 64, 1326-1329, (1990).
    • (1990) Phys. Rev. Lett. , vol.64 , pp. 1326-1329
    • Periwal, V.1    Shevitz, D.2
  • 48
    • 10044262136 scopus 로고    scopus 로고
    • Increasing subsequences and the classical groups
    • MR 98k:05146
    • E.M.Rains, Increasing subsequences and the classical groups, Electron. J. of Combinatorics, 5(1), R12, (1998). MR 98k:05146
    • (1998) Electron. J. of Combinatorics , vol.5 , Issue.1
    • Rains, E.M.1
  • 51
    • 0001259916 scopus 로고
    • Longest increasing and decreasing subsequences
    • MR 22:12047
    • C.Schensted, Longest increasing and decreasing subsequences, Canad. J. Math., 13, 179 - 191, (1961). MR 22:12047
    • (1961) Canad. J. Math. , vol.13 , pp. 179-191
    • Schensted, C.1
  • 52
    • 3042554249 scopus 로고    scopus 로고
    • A microscopic model for the burgers equation and longest increasing subsequences
    • MR 97d:60162
    • T.Seppäläinen, A microscopic model for the Burgers equation and longest increasing subsequences, Electron. J. Prob., 1, no.5, (1996) MR 97d:60162
    • (1996) Electron. J. Prob. , vol.1 , Issue.5
    • Seppäläinen, T.1
  • 53
    • 0032188524 scopus 로고    scopus 로고
    • Large deviations for increasing sequences on the plane
    • CMP 99:04
    • T.Seppäläinen, Large deviations for increasing sequences on the plane, Probab. Theory Related Fields, 112, no.2, 221-244, (1998). CMP 99:04
    • (1998) Probab. Theory Related Fields , vol.112 , Issue.2 , pp. 221-244
    • Seppäläinen, T.1
  • 54
    • 0003320835 scopus 로고    scopus 로고
    • Representations of finite and compact groups
    • American Mathematical Society, MR 97c:22001
    • B.Simon, Representations of Finite and Compact Groups, Graduate Studies in Mathematics vol. 10, American Mathematical Society, 1996. MR 97c:22001
    • (1996) Graduate Studies in Mathematics , vol.10
    • Simon, B.1
  • 56
    • 0038887876 scopus 로고
    • On certain hermitian forms associated with the fourier series of a positive function
    • 228-237, (1952) Gabor Szego: Collected Papers - (19451972), Birkhäuser, MR 14:553d
    • G.Szegö, On Certain Hermitian Forms Associated with the Fourier Series of a Positive Function, Comm. Seminaire Math de l'Univ. de Lund, tome supplementaire, dedie a Marcel Riesz, 228-237, (1952) (or Gabor Szego: Collected Papers - Vol 3 (1945-1972), 270-280, Birkhäuser, 1982). MR 14:553d
    • (1982) Comm. Seminaire Math de L'univ. de Lund, Tome Supplementaire, Dedie a Marcel Riesz , vol.3 , pp. 270-280
  • 57
    • 27644532196 scopus 로고
    • Level-spacing distributions and the Airy kernel
    • MR 95e:82003
    • C.A.Tracy and H.Widom, Level-Spacing distributions and the Airy kernel, Comm. Math. Phys., 159, 151-174, (1994). MR 95e:82003
    • (1994) Comm. Math. Phys. , vol.159 , pp. 151-174
    • Tracy, C.A.1    Widom, H.2
  • 60
    • 0000057952 scopus 로고
    • Asymptotics of the plancherel measure of the symmetric group and the limiting form of young tables
    • MR 58:562
    • A.M.Vershik and S.V.Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables, Soviet Math. Dokl., 18, 527-531, (1977). MR 58:562
    • (1977) Soviet Math. Dokl. , vol.18 , pp. 527-531
    • Vershik, A.M.1    Kerov, S.V.2
  • 61
    • 34250123019 scopus 로고
    • Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group
    • MR 86k:11051
    • A.M.Vershik and S.V.Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Functional Anal. Appl., 19, no.1, 21-31, (1985). MR 86k:11051
    • (1985) Functional Anal. Appl. , vol.19 , Issue.1 , pp. 21-31
    • Vershik, A.M.1    Kerov, S.V.2
  • 62
    • 0040072344 scopus 로고    scopus 로고
    • personal communication
    • H.Widom, personal communication.
    • Widom, H.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.