Aspects of large random Markov kernels

Stochastics

Volumn 81, Issue 3-4, 2009, Pages 415-429

  Chafaï, Djalil ^a  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

Convex polytopes; Finite graphs; Invariant measures; Markov chains; Random matrices; Symmetric group

Indexed keywords

EID: 70450193509     PISSN: 17442508     EISSN: 17442516     Source Type: Journal    
DOI: 10.1080/17442500903080314     Document Type: Article

References (46)

1. R. Arratia, A.D. Barbour, and S. Tavaré, Poisson process approximations for the Ewens sampling formula, Ann. Appl. Probab. 2(3) (1992), pp. 519-535.
2. p, Positivity 10(4) (2006), pp. 755-759.
3. A. Auffinger, G. Ben Arous, and S. Péché, Poisson convergence for the largest eigenvalues of Heavy Tailed Random Matrices, 2007, preprint, arXiv:0710.3132 [math.PR].
4. Z.D. Bai and Y.Q. Yin, Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix, Ann. Probab. 21(3) (1993), pp. 1275-1294.
5. n ball, Ann. Probab. 33(2) (2005), pp. 480-513, arXiv.org:math.PR/0503650.
6. G. Ben Arous and A. Guionnet, The spectrum of heavy-tailed random matrices, 2007, preprint arXiv:0707.2159 [math.PR], July.
7. I. Bengtsson, Å. Ericsson, M. Kú, W. Tadej, and K. Życzkowski, Birkhoff's polytope and unistochastic matrices, N = 3 and N = 4, Commun. Math. Phys. 259(2) (2005), pp. 307-324.
8. G. Biroli, J.-P. Bouchaud, and M. Potters, On the top eigenvalue of heavy-tailed random matrices, Europhys. Lett. EPL 78(1) (2007), Art. 10001, 5.
9. d in a random environment, Stochastic Process. Appl. 104(1) (2003), pp. 29-56.
10. B. Bollobás, Random Graphs, 2nd ed., Cambridge Studies in Advanced Mathematics, Vol. 73, Cambridge University Press, Cambridge, 2001.
11. Ch. Bordenave, P. Caputo, and D. Chafaï, Spectrum of large random reversible Markov chains, 2008, arXiv:0811.1097 [math.PR].
12. Ch. Bordenave, P. Caputo, and D. Chafaï, Spectrum of large random reversible Markov chains - heavy-tailed weights on the complete graph, 2009, in preparation.
13. J.-P. Bouchaud, A. Georges, and P. Le Doussal, Diffusion anormale dans les milieux désordonnés: piégeage, corrélations et théorèmes de la limite centrale, J. Phys. 48(11) (1987), pp. 1855-1860.
14. A. Bovier and A. Faggionato, Spectral characterization of aging: The REM-like trap model, Ann. Appl. Probab. 15(3) (2005), pp. 1997-2037.
15. A. Bovier and A. Faggionato, Spectral analysis of Sinai's walk for small eigenvalues, Ann. Probab. 36(1) (2008), pp. 198-254.
16. S. Boyd, P. Diaconis, P. Parrilo, and L. Xiao, Symmetry analysis of reversible Markov chains, Internet Math. 2(1) (2005), pp. 31-71.
17. W. Bryc, A. Dembo, and T. Jiang, Spectral measure of large random Hankel, Markov and Toeplitz matrices, Ann. Probab. 34(1) (2006), pp. 1-38.
18. M. Capitaine, C. Donati-Martin, and D. Féral, The largest eigenvalue of finite rank deformation of large Wigner matrices: Convergence and non-universality of the fluctuations, Ann. Probab. 2007, to appear.
19. D. Chafaï, The Dirichlet Markov Ensemble, 2007, preprint arXiv:0709.4678 [math.PR].
20. D. Chafaï, Circular law theorem for random Markov matrices, 2008, notes arXiv:0808.1502 [math.PR].
21. J.A. De Loera, F. Liu, and R. Yoshida, Formulas for the volumes of the polytope of doubly-stochastic matrices and its faces, 2007, preprint arXiv:math/0701866 [math.CO].
22. D. Féral and S. Péché, The largest eigenvalue of rank one deformation of large Wigner matrices, Commun. Math. Phys. 272(1) (2007), pp. 185-228.
23. G. Goldberg and M. Neumann, Distribution of subdominant eigenvalues of matrices with random rows, SIAM J. Matrix Anal. Appl. 24(3) (2003), pp. 747-761 (electronic).
24. G. Goldberg, P. Okunev, M. Neumann, and H. Schneider, Distribution of subdominant eigenvalues of random matrices, Methodol. Comput. Appl. Probab. 2(2) (2000), pp. 137-151.
25. B.M. Hambly, P. Keevash, N. O'Connell, and D. Stark, The characteristic polynomial of a random permutation matrix, Stochastic Process. Appl. 90(2) (2000), pp. 335-346.
26. R.A. Horn and Ch.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990, Corrected reprint of the 1985 original.
27. J.B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Determinantal processes and independence, Probab. Surv. 3 (2006), pp. 206-229 (electronic).
28. F.P. Kelly, Reversibility and Stochastic Networks, Wiley, Chichester, 1979, Wiley Series in Probability and Mathematical Statistics.
29. E. Kostlan, On the spectra of Gaussian matrices, Linear Algebra Appl. 162/164 (1992), pp. 385-388, Directions in matrix theory (Auburn, AL, 1990).
30. M. Marcus and R. Ree, Diagonals of doubly stochastic matrices, Quart. J. Math. Oxford Ser. 10(2) (1959), pp. 296-302.
31. L. Mirsky, Results and problems in the theory of doubly-stochastic matrices, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1 (1962/1963), pp. 319-334.
32. G. Pan and W. Zhou, Circular law, extreme singular values and potential theory, 2007, arXiv:0705.3773 [math.PR].
33. S. Péché, The largest eigenvalue of small rank perturbations of Hermitian random matrices, Probab. Theor. Relat. Fields 134(1) (2006), pp. 127-173.
34. I. Popescu, General tridiagonal random matrix models, limiting distributions and fluctuations, Probab. Theor. Relat. Fields (2008), arXiv:math/0610827 [math.PR], to appear.
35. B. Rider, A limit theorem at the edge of a non-Hermitian random matrix ensemble, J. Phys. A 36(12) (2003), pp. 3401-3409, Random matrix theory.
36. B. Rider, Order statistics and Ginibre's ensembles, J. Statist. Phys. 114(3-4) (2004), pp. 1139-1148.
37. M. Rosenblatt, Markov Processes. Structure and Asymptotic Behavior, Springer, New York, NY, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 184.
38. M. Rudelson and R. Vershynin, The Littlewood-Offord Problem and invertibility of random matrices, 2007, preprint arXiv:math/0703503 [math.PR].
39. L. Saloff-Coste, Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996), Vol. 1665, Lecture Notes in Math., Springer, Berlin, 1997, pp. 301-413.
40. E. Seneta, Non-negative matrices and Markov chains, Springer Series in Statistics, Springer, New York, 2006, Revised reprint of the second (1981) edition.
41. A. Soshnikov, Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207(3) (1999), pp. 697-733.
42. A. Soshnikov, Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails, Electron. Comm. Probab. 9 (2004), pp. 82-91 (electronic).
43. G. Tanner, Unitary-stochastic matrix ensembles and spectral statistics, J. Phys. A 34(41) (2001), pp. 8485-8500.
44. T. Tao and V. Vu, Random matrices: The circular law, 2007, preprint arXiv:0708.2895 [math.PR].
45. C.A. Tracy and H. Widom, The distribution of the largest eigenvalue in the Gaussian ensembles: β = 1,2,4, Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, NY, 2000, pp. 461-472.
46. K. Wieand, Eigenvalue distributions of random permutation matrices, Ann. Probab. 28(4) (2000), pp. 1563-1587.