메뉴 건너뛰기




Volumn 17, Issue 2, 2008, Pages 231-244

The distribution of the largest nontrivial eigenvalues in families of random regular graphs

Author keywords

Largest nontrivial eigenvalues; Ramanujan graphs; Random graphs; Tracy Widom distribution

Indexed keywords


EID: 50049102718     PISSN: 10586458     EISSN: 1944950X     Source Type: Journal    
DOI: 10.1080/10586458.2008.10129029     Document Type: Article
Times cited : (38)

References (46)
  • 1
    • 0034336152 scopus 로고    scopus 로고
    • The Statistical Theory of Quantum Dots
    • Y. Alhassid. “The Statistical Theory of Quantum Dots.” Rev. Mod. Phys. 72: 4 (2000), 895–968.
    • (2000) Rev. Mod. Phys , vol.72 , Issue.4 , pp. 895-968
    • Alhassid, Y.1
  • 2
    • 51249173340 scopus 로고
    • Eigenvalues and Expanders
    • N. Alon. “Eigenvalues and Expanders.” Combinatorica 6: 2 (1986), 83–96.
    • (1986) Combinatorica , vol.6 , Issue.2 , pp. 83-96
    • Alon, N.1
  • 3
    • 33750994100 scopus 로고
    • λ1, Isoperimetric Inequalities for Graphs, and Superconcentrators
    • N. Alon and V. Milman. “λ1, Isoperimetric Inequalities for Graphs, and Superconcentrators.” J. Combin. Theory Ser. B 38: 1 (1985), 73–88.
    • (1985) J. Combin. Theory Ser. B , vol.38 , Issue.1 , pp. 73-88
    • Alon, N.1    Milman, V.2
  • 4
    • 0034338413 scopus 로고    scopus 로고
    • Limiting distributions for a polynuclear growth model
    • J. Baik and E. M. Rains. Limiting distributions for a polynuclear growth model, J. Stat. Phys. 100 (2000), 523–541.
    • (2000) J. Stat. Phys , vol.100 , pp. 523-541
    • Baik, J.1    Rains, E.M.2
  • 5
    • 0010151103 scopus 로고    scopus 로고
    • The Asymptotics of Monotone Subsequences of Involutions
    • J. Baik and E. M. Rains. “The Asymptotics of Monotone Subsequences of Involutions.” Duke Math. J. 109 (2001), 205–281.
    • (2001) Duke Math. J , vol.109 , pp. 205-281
    • Baik, J.1    Rains, E.M.2
  • 6
    • 0010126887 scopus 로고    scopus 로고
    • Symmetrized Random Permutations
    • Random Matrix Models and Their Applications, P. Bleher and A. Its, Cambridge: Cambridge University Press
    • J. Baik and E. M. Rains. “Symmetrized Random Permutations.” In Random Matrix Models and Their Applications, edited by P. Bleher and A. Its, pp. 1–19, Math. Sci. Res. Inst. Publications 40. Cambridge: Cambridge University Press, 2001.
    • (2001) Math. Sci. Res. Inst. Publications , pp. 1-19
    • Baik, J.1    Rains, E.M.2
  • 7
    • 0033443150 scopus 로고    scopus 로고
    • On the Distribution of the Length of the Longest Increasing Subsequence of Random Permutations
    • 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. 12 (1999), 1119–1178.
    • (1999) J. Amer. Math. Soc , vol.12 , pp. 1119-1178
    • Baik, J.1    Deift, P.2    Johansson, K.3
  • 9
    • 0005688259 scopus 로고
    • Constructions of Telephone Networks by Group Representations
    • F. Bien. “Constructions of Telephone Networks by Group Representations.” Notices of the AMS 36: 1 (1989), 5–22.
    • (1989) Notices of the AMS , vol.36 , Issue.1 , pp. 5-22
    • Bien, F.1
  • 10
    • 11944253901 scopus 로고    scopus 로고
    • Cambridge: Cambridge University Press
    • B. Bollobás. Random Graphs. Cambridge: Cambridge University Press, 2001.
    • (2001) Random Graphs
    • Bollobás, B.1
  • 11
    • 0034390399 scopus 로고    scopus 로고
    • Asymptotics of Plancherel Measures for Symmetric Groups
    • A. Borodin, A. Okounkov, and G. Olshanski. “Asymptotics of Plancherel Measures for Symmetric Groups.” J. Amer. Math. Soc. 13 (2000), 481–515.
    • (2000) J. Amer. Math. Soc , vol.13 , pp. 481-515
    • Borodin, A.1    Okounkov, A.2    Olshanski, G.3
  • 12
    • 33749564809 scopus 로고
    • Cubic Ramanujan Graphs
    • P. Chiu. “Cubic Ramanujan Graphs. Combinatorica 12: 3 (1992), 275–285.
    • (1992) Combinatorica , vol.12 , Issue.3 , pp. 275-285
    • Chiu, P.1
  • 13
    • 0012526216 scopus 로고    scopus 로고
    • Elementary Number Theory, Group Theory, and Ramanujan Graphs
    • London Mathematical Society, Student Texts 55. Cambridge: Cambridge University Press
    • G. Davidoff, P. Sarnak, and A. Valette. Elementary Number Theory, Group Theory, and Ramanujan Graphs, London Mathematical Society, Student Texts 55. Cambridge: Cambridge University Press, 2003.
    • (2003)
    • Davidoff, G.1    Sarnak, P.2    Valette, A.3
  • 14
    • 0001398283 scopus 로고
    • Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks
    • J. Dodziuk. “Difference Equations, Isoperimetric Inequality and Transience of Certain Random Walks.” Trans. Amer. Math. Soc. 284: 2 (1984), 787–794.
    • (1984) Trans. Amer. Math. Soc , vol.284 , Issue.2 , pp. 787-794
    • Dodziuk, J.1
  • 15
    • 84974003380 scopus 로고
    • Some Geometric Aspects of Graphs and Their Eigenfunctions
    • J. Friedman. “Some Geometric Aspects of Graphs and Their Eigenfunctions.” Duke Math. J. 69: 3 (1993), 487–525.
    • (1993) Duke Math. J , vol.69 , Issue.3 , pp. 487-525
    • Friedman, J.1
  • 17
    • 0001865474 scopus 로고
    • Sur la loi limite de lespacement des valeurs propres d’une matrice al´eatoire
    • M. Gaudin. “Sur la loi limite de l’espacement des valeurs propres d’une matrice al´eatoire.” Nucl. Phys. 25 (1961), 447–458.
    • (1961) Nucl. Phys , vol.25 , pp. 447-458
    • Gaudin, M.1
  • 19
    • 0035534219 scopus 로고    scopus 로고
    • Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models
    • J. Gravner, C. A. Tracy, and H. Widom, “Limit Theorems for Height Fluctuations in a Class of Discrete Space and Time Growth Models.” J. Stat. Phys. 102 (2001), 1085–1132.
    • (2001) J. Stat. Phys , vol.102 , pp. 1085-1132
    • Gravner, J.1    Tracy, C.A.2    Widom, H.3
  • 20
  • 21
    • 0242353284 scopus 로고    scopus 로고
    • Eigenvalue Spacings for Regular Graphs
    • (Minneapolis, 1996), IMA Volumes in Mathematics and Its Applications 109. New York: Springer
    • D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick. “Eigenvalue Spacings for Regular Graphs.” In Emerging Applications of Number Theory (Minneapolis, 1996), pp. 317–327, IMA Volumes in Mathematics and Its Applications 109. New York: Springer, 1999.
    • (1999) Emerging Applications of Number Theory , pp. 317-327
    • Jakobson, D.1    Miller, S.D.2    Rivin, I.3    Rudnick, Z.4
  • 22
    • 0035626398 scopus 로고    scopus 로고
    • Discrete Orthogonal Polynomial Ensembles and the Plancherel Measure
    • K. Johansson. “Discrete Orthogonal Polynomial Ensembles and the Plancherel Measure.” Ann. Math. 153 (2001), 259–296.
    • (2001) Ann. Math , vol.153 , pp. 259-296
    • Johansson, K.1
  • 23
    • 0036628218 scopus 로고    scopus 로고
    • Non-intersecting Paths, Random Tilings and Random Matrices
    • K. Johansson. “Non-intersecting Paths, Random Tilings and Random Matrices.” Probab. Th. Rel. Fields 123 (2002), 225–280.
    • (2002) Probab. Th. Rel. Fields , vol.123 , pp. 225-280
    • Johansson, K.1
  • 27
    • 0003059022 scopus 로고
    • Explicit Group-Theoretic Constructions of Combinatorial Schemes and Their Applications in the Construction of Expanders and Concentrators
    • translation in Problems Inform. Transmission 24: 1 (1988), 39–46
    • G. A. Margulis. “Explicit Group-Theoretic Constructions of Combinatorial Schemes and Their Applications in the Construction of Expanders and Concentrators” (Russian). Problemy Peredachi Informatsii 24: 1 (1988), 51–60; translation in Problems Inform. Transmission 24: 1 (1988), 39–46.
    • (1988) Problemy Peredachi Informatsii , vol.24 , Issue.1 , pp. 51-60
    • Margulis, G.A.1
  • 28
    • 0000294458 scopus 로고
    • The Expected Eigenvalue Distribution of a Large Regular Graph
    • B. McKay. “The Expected Eigenvalue Distribution of a Large Regular Graph.” Linear Algebra Appl. 40 (1981), 203–216.
    • (1981) Linear Algebra Appl , vol.40 , pp. 203-216
    • McKay, B.1
  • 29
    • 0001074948 scopus 로고
    • On the Statistical Properties of Level Spacings in Nuclear Spectra
    • M. Mehta. “On the Statistical Properties of Level Spacings in Nuclear Spectra.” Nucl. Phys. 18 (1960), 395-419.
    • (1960) Nucl. Phys , vol.18 , pp. 395-419
    • Mehta, M.1
  • 30
    • 0004163930 scopus 로고
    • 2nd edition. Boston: Academic Press
    • M. Mehta. Random Matrices, 2nd edition. Boston: Academic Press, 1991.
    • (1991) Random Matrices
    • Mehta, M.1
  • 31
    • 0002622843 scopus 로고
    • Existence and Explicit Constructions of q +1 Regular Ramanujan Graphs for Every Prime Power q
    • M. Morgenstern. “Existence and Explicit Constructions of q +1 Regular Ramanujan Graphs for Every Prime Power q.” J. Combin. Theory Ser. B 62: 1 (1994), 44–62.
    • (1994) J. Combin. Theory Ser. B , vol.62 , Issue.1 , pp. 44-62
    • Morgenstern, M.1
  • 32
    • 33646775825 scopus 로고    scopus 로고
    • Ramanujan Graphs
    • M. Ram Murty, “Ramanujan Graphs.” J. Ramanujan Math. Soc. 18: 1 (2003), 33–52.
    • (2003) J. Ramanujan Math. Soc , vol.18 , Issue.1 , pp. 33-52
    • Ram Murty, M.1
  • 33
    • 84968518845 scopus 로고
    • On the Distribution of Spacings between Zeros of the Zeta Function
    • A. Odlyzko. “On the Distribution of Spacings between Zeros of the Zeta Function.” Math. Comp. 48: 177 (1987), 273–308.
    • (1987) Math. Comp , vol.48 , Issue.177 , pp. 273-308
    • Odlyzko, A.1
  • 34
    • 0041335000 scopus 로고    scopus 로고
    • The 1022-nd Zero of the Riemann Zeta Function
    • edited by M. van Frankenhuysen and M. L. Lapidus, Amer. Math. Soc., Contemporary Math. Series. Available online
    • A. Odlyzko. “The 1022-nd Zero of the Riemann Zeta Function.” In Proc. Conference on Dynamical, Spectral and Arithmetic Zeta-Functions, edited by M. van Frankenhuysen and M. L. Lapidus, Amer. Math. Soc., Contemporary Math. Series. Available online (http://www.research.att.com/∼amo/doc/zeta.html), 2001.
    • (2001) Proc. Conference on Dynamical, Spectral and Arithmetic Zeta-Functions
    • Odlyzko, A.1
  • 35
    • 0001381109 scopus 로고
    • Super Concentrators
    • N. Pippenger. “Super Concentrators.” SIAM Journal Comp. 6 (1977), 298–304.
    • (1977) SIAM Journal Comp , vol.6 , pp. 298-304
    • Pippenger, N.1
  • 36
    • 0343391145 scopus 로고    scopus 로고
    • Statistical Self-Similarity of One-Dimensional Growth Processes
    • M. Prähofer and H. Spohn. “Statistical Self-Similarity of One-Dimensional Growth Processes.” Physica A 279 (2000), 342–352.
    • (2000) Physica A , vol.279 , pp. 342-352
    • Prähofer, M.1    Spohn, H.2
  • 37
    • 0000439075 scopus 로고    scopus 로고
    • Universal Distributions for Growth Processes in 1 + 1 Dimensions and Random Matrices
    • M. Prähofer and H. Spohn. “Universal Distributions for Growth Processes in 1 + 1 Dimensions and Random Matrices.” Phys. Rev. Lett. 84 (2000), 4882–4885.
    • (2000) Phys. Rev. Lett , vol.84 , pp. 4882-4885
    • Prähofer, M.1    Spohn, H.2
  • 38
    • 0003288907 scopus 로고
    • Some Applications of Modular Forms
    • Cambridge: Cambridge University Press
    • P. Sarnak. Some Applications of Modular Forms, Cambridge Trusts in Mathematics, 99, Cambridge: Cambridge University Press, 1990.
    • (1990) Cambridge Trusts in Mathematics , vol.99
    • Sarnak, P.1
  • 39
    • 23444458006 scopus 로고    scopus 로고
    • What Is an Expander?
    • P. Sarnak. “What Is an Expander?” Notices of the AMS 51: 7 (2004), 762–763.
    • (2004) Notices of the AMS , vol.51 , Issue.7 , pp. 762-763
    • Sarnak, P.1
  • 41
    • 0030290419 scopus 로고    scopus 로고
    • Expander Codes
    • M. Sipser and Spielman, “Expander Codes.” IEEE Trans. Inform. Theory 42: 6 (1996), 1710–1722.
    • (1996) IEEE Trans. Inform. Theory , vol.42 , Issue.6 , pp. 1710-1722
    • Sipser, M.1
  • 42
    • 27644532196 scopus 로고
    • Level-Spacing Distributions and the Airy Kernel
    • C. A. Tracy and H. Widom. “Level-Spacing Distributions and the Airy Kernel.” Comm. Math. Phys. 159 (1994), 151–174.
    • (1994) Comm. Math. Phys , vol.159 , pp. 151-174
    • Tracy, C.A.1    Widom, H.2
  • 43
    • 0030545965 scopus 로고    scopus 로고
    • Widom. “On Orthogonal and Sympletic Matrix Ensembles
    • C. Tracy and H. Widom. “On Orthogonal and Sympletic Matrix Ensembles. Communications in Mathematical Physics 177 (1996), 727–754.
    • (1996) Communications in Mathematical Physics , vol.177 , pp. 727-754
    • Tracy, C.1
  • 44
    • 3042622312 scopus 로고    scopus 로고
    • Distribution Functions for Largest Eigenvalues and Their Applications
    • C. Tracy and H. Widom. “Distribution Functions for Largest Eigenvalues and Their Applications.” ICM Vol. I (2002), 587–596.
    • (2002) ICM , vol.1 , pp. 587-596
    • Tracy, C.1    Widom, H.2
  • 45
    • 0035137780 scopus 로고    scopus 로고
    • Universal Gap Fluctuations in the Superconductor Proximity Effect
    • M. G. Vavilov, P. W. Brouwer, V. Ambegaokar, and C. W. J. Beenakker. “Universal Gap Fluctuations in the Superconductor Proximity Effect.” Phys. Rev. Lett. 86 (2001), 874–877.
    • (2001) Phys. Rev. Lett , vol.86 , pp. 874-877
    • Vavilov, M.G.1    Brouwer, P.W.2    Ambegaokar, V.3    Beenakker, C.W.J.4
  • 46
    • 0003013212 scopus 로고
    • Statistical Properties of Real Symmetric Matrices
    • Toronto: University of Toronto Press
    • E. Wigner. “Statistical Properties of Real Symmetric Matrices.” In Canadian Mathematical Congress Proceedings, pp. 174–184. Toronto: University of Toronto Press, 1957.
    • (1957) Canadian Mathematical Congress Proceedings , pp. 174-184
    • Wigner, E.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.