Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices

By Percy Deift, Dimitri Gioev. 2005

157 citations recorded by the source corpus.

We prove universality at the edge of the spectrum for unitary (β = 2), orthogonal (β = 1), and symplectic (β = 4) ensembles of random matrices in the scaling limit for a class of weights w(x) = e−V(x) where V is a polynomial, V(x) = κ2mx2m + · · ·, κ2m > 0. The precise statement of our results is given in Theorem 1.1 and Corollaries 1.2 and 1.4 below. For the same class of weights, a proof of universality in the bulk of the spectrum is given in [12] for the unitary ensembles and in [9] for the orthogonal and symplectic ensembles.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e9f4e9cd512c58488493307c74d3bdbfac533622

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Citations in the corpus

A Dynamical Approach to Random Matrix Theory (2017, methodology): “Similar to the bulk case, the edge universality was first proved via orthogonal polynomial methods for the classical values of β = 1, 2, 4 in [38,42,110,116].”
Sparse regular random graphs: Spectral density and eigenvectors (2009, background): “Deift and Goiev [19] looked at different potential functions on symmetric, hermitian, and self-dual matrices; Baik and Suidan explored connections to percolation [7] and random walks [8]; β-generalizations of the classical ensembles and universal properties thereof have been explored in Forrester an”
Universality of the Stochastic Airy Operator (2013, background): “In this case V is called regular and the rescaled maximal eigenvalue has the appropriate Tracy-Widom law (see [11], again for ˇ D 1; 2; 4).”
Universality for mathematical and physical systems (2006, unknown): “al edge E = λmax, one again ﬁnds universal behavior for UE’s with potentials VN,2, in particular, of the form (23) above. For such VN,2’s there exist constants z(2) N ,s (2) N such that for t ∈R (see [DeiGio2] and the notes therein) lim N→∞ Prob M : λmax −z (2) N s(2) N ≤t ! = det 1−A↾L2(t,∞) . (30)”
Edge Universality of Beta Ensembles (2013, background): “The case β = 2 and real analytic V was solved in [12, 15].”
The Distributions of Random Matrix Theory and their Applications (2009, result): “The results for the unitary case (β = 2) are due to Deift, Kriecherbaur, McLaughlin, Venakides and Zhou [9] and the orthogonal/symplectic results are recent work of Deift and Gioev [8].”
Higher‐order analogues of the Tracy‐Widom distribution and the Painlevé II hierarchy (2009, background): “In the generic case where k = 0 for the rightmost endpoint b = bV of suppψV , it follows from the results of Deift et al [20, 16] that, for any fixed u and v, lim n→∞ 1 cn2/3 Kn(b+ u cn2/3 , b+ v cn2/3 ) = K(0)(u, v), (1.6) where K(0)(u, v) = Ai (u)Ai ′(v) − Ai (v)Ai ′(u) u− v (1.7) is the Airy kern”
Rank 1 real Wishart spiked model (2011, methodology): “To obtain the results for the 21 entry, we use the fact that ǫ (S1) (x, y) is skew symmetric to obtain (See [31], [32], [33]) ǫ (S1) (x, y) = ǫ (S1) (x, y)− ǫ (S1) (y, y) = − ∫ y x S1(t, y)dt.”
The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights (2008, methodology): “Asymptotic Result 2 was established for general real analytic potentials V in [7], and the asymptotic formulae for orthogonal polynomials given in [7] were used to establish Asymptotic Result 1 in [5].”