Parameter Symmetry in Perturbed GUE Corners Process and Reflected Drifted Brownian Motions
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Parameter Symmetry in Perturbed GUE Corners Process and Reflected Drifted Brownian Motions Leonid Petrov1,2 · Mikhail Tikhonov2,3 Received: 24 January 2020 / Accepted: 3 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The perturbed GUE corners ensemble is the joint distribution of eigenvalues of all principal submatrices of a matrix G +diag(a), where G is the random matrix from the Gaussian Unitary Ensemble (GUE), and diag(a) is a fixed diagonal matrix. We introduce Markov transitions based on exponential jumps of eigenvalues, and show that their successive application is equivalent in distribution to a deterministic shift of the matrix. This result also leads to a new distributional symmetry for a family of reflected Brownian motions with drifts coming from an arithmetic progression. The construction we present may be viewed as a random matrix analogue of the recent results of the first author and Axel Saenz [17]. Keywords Random matrices · Perturbed GUE corner process · Reflected Brownian motions
1 Introduction 1.1 Couplings for Perturbed GUE Corners Process The Gaussian Unitary Ensemble (GUE) is the most well-known random matrix model [2,6, 14]. This paper presents a new symmetry of the distribution of the perturbed GUE ensemble. By this we mean the random matrix ensemble of the form H = G + diag(a1 , . . . , a N ), where G is an N × N GUE random matrix, to which we add a fixed diagonal matrix. This model is often also called GUE with external source. We refer to [3–5] and references therein for the history of the perturbed ensemble and various asymptotic results. (In fact, below we consider
Communicated by Antti Knowles.
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Mikhail Tikhonov [email protected] ; [email protected] Leonid Petrov [email protected]
1
Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA
2
Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, Russia 127994
3
Faculty of Physics, Lomonosov Moscow State University, Leninskie Gory, 1-2, Moscow, Russia 119991
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L. Petrov, M. Tikhonov
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Fig. 1 Interlacing array of eigenvalues of all principal corners of a 4 × 4 matrix
a slightly more general version of the matrix model involving a time-dependent rescaling; this version is suitable for the application to reflected Brownian motions). The unperturbed GUE random matrix, corresponding to ai ≡ 0, is unitary invariant in d
the sense that there is equality in distribution G = U GU ∗ for any fixed N × N unitary matrix U . This implies that the distribution of the eigenvalues of H is symmetric in the perturbation parameters a1 , . . . , a N . The overall goal of the paper is to explore probabilistic consequences of this symmetry property. N N Together with the eigenvalues λ N = (λ N N ≤ . . . ≤ λ1 ), λi ∈ R, of the full matrix N 1 H = [h i j ]i, j=1 , one can also consider its corners proces
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