Aspects of large random Markov kernels

References

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