Matrix models for circular ensembles

By Rowan Killip, I. Nenciu. 2004

248 citations recorded by the source corpus.

The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by

Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/13dd78831e2c888f7b2ce503f92bd7f8b412065f

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In Condensed Matter Physics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

OPUC on one foot (2005, background): “In this regard, there is a natural question answered by Killip-Nenciu [50].”
CMV matrices: Five years after (2006, methodology): “The normalization is the inverse of the beta function 22−n (n−1)/ ( n−1 2 ) which, as noted by [50] (there is a ( )−1 missing on the leftmost term in their (3.”
Eigenvalue statistics for CMV matrices: From Poisson to clock via random matrix ensembles (2009, background): “(ii) If E { |αk|2 } − 2β(k+1) = o(k −1) and E { log2[1− |αk|2] } = o(k−1), then the eigenvalue process converges to the β-ensemble as n→∞.”
The Analytic Theory of Matrix Orthogonal Polynomials (2007, methodology): “There are four proofs that we know of: the original argument of Geronimus [90] based on Szegő’s formula in [182], a proof of Damanik–Killip [32] using Schur functions, a proof of Killip–Nenciu [127] using CMV matrices, and a proof of Faybusovich–Gekhtman [81] using canonical moments.”
Monte Carlo with determinantal point processes (2016, background): “18), sampling the OP Ensemble can already be done rejection-free and in time O(N2) by diagonalizing a tridiagonal random matrix that only requires sampling independent beta variables [Killip and Nenciu, 2004].”
FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS, I. A TALE OF TWO PICTURES (2004, background): “Since any unitary with distinguished cyclic vector can be represented by a CMV matrix, CUE has a realization connected with OPUC, just not either the totally random or BLS case. Indeed, Killip-Nenciu [19] have shown that CUE is given by independent αj’s but not identically distributed. In Section 3,”
Asymptotic behavior of random determinants in the Laguerre, Gram and Jacobi ensembles (2006, background): “Since the construction ofthe bbi isrecursive,itispossible (for xed n)to considerthe whole sequence ofvariables fdetX n;r;r= 1;:::;ng atthe sam etim e. Itcorrespondsto the decom position ofthe determ inantofa sym m etricpositive m atrix A as detA = rY j= 1 detA [j] detA [j 1] ; where A [j] is the j…”
The importance of the Selberg integral (2007, methodology): “Nenciu [92], in a study which does not make use of the Dixon–Anderson integral (2.”
GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES (2012, background): “For an extensive study of the β = 1 case, as well as a clear exposition of how these models arose, we refer to [36]; the other cases (β = 2, 4) can be dealt with similarly.”