What is the probability that a large random matrix has no real eigenvalues?

Kanzieper, Eugene; Poplavskyi, Mihail; Timm, Carsten; Tribe, Roger; Zaboronski, Oleg

doi:10.1214/15-aap1160

Cited by 19 publications

(27 citation statements)

References 13 publications

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“…This latter result is consistent with the exact result p P 1 N,N = 2 −N(N−1)/4 for P 1 a real standard Gaussian matrix [8]. For P 1 a real standard Gaussian matrix, the probability p P 1 N,0 (with N even) that all eigenvalues are complex has the large N expansion [23,13]…”

Section: )supporting

confidence: 83%

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

Forrester

Ipsen

Kumar

2018

Experimental Mathematics

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A truncation of a Haar distributed orthogonal random matrix gives rise to a matrix whose eigenvalues are either real or complex conjugate pairs, and are supported within the closed unit disk. This is also true for a product P m of m independent truncated orthogonal random matrices. One of most basic questions for such asymmetric matrices is to ask for the number of real eigenvalues. In this paper, we will exploit the fact that the eigenvalues of P m form a Pfaffian point process to obtain an explicit determinant expression for the probability of finding any given number of real eigenvalues. We will see that if the truncation removes an even number of rows and columns from the original Haar distributed orthogonal matrix, then these probabilities will be rational numbers. Finally, based on exact finite formulae, we will provide conjectural expressions for the asymptotic form of the spectral density and the average number of real eigenvalues as the matrix dimension tends to infinity.

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Section: )supporting

confidence: 83%

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

Forrester

Ipsen

Kumar

2018

Experimental Mathematics

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“…A step in this direction was taken in [20], where using the relation to the Brownian annihilation process A + A → ∅, the first two terms of the large s asymptotics of the probability that there are no real eigenvalues in an interval of size s near the origin for N → ∞ real Ginibre (m = 1) was computed. It was realized Kanzieper et al [34] that heuristic at least this result implies for large N…”

Section: Probability Of K Real Eigenvaluesmentioning

confidence: 92%

“…≈ 0.0627, (3.25) and moreover these authors gave a rigorous proof of the leading term. It is not known how to generalize the workings of [34], which are based on Theorem 1, beyond m = 1. However, our Theorem 1 at least allows us to establish numerical estimates, e.g.…”

Section: Probability Of K Real Eigenvaluesmentioning

confidence: 99%

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Forrester

Ipsen

2016

Linear Algebra and its Applications

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Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these -the eigenvalue density of the real eigenvalues -gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

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“…Using the statistical independence of the positive and negative real eigenvalues for large n, one has Q 0 (x, n) ∼ [Q 0 (x, n)] 2 , and in particular Q 0 (1, n) ∼ n −2b for large n. Using similar arguments, one can show that the full statistics of the zero-crossings of the diffusion equation (equivalently of the real roots of K n (x)) can be obtained, at leading order for large n, from the statistics of the number of real eigenvalues N n of the random matrix M 2n , which we now study. Our analysis follows the line developed in [46] where the real eigenvalues of real Ginibre matrices were studied. We start with the full joint distribution of the eigenvalues of M 2n (8).…”

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confidence: 99%

Exact Persistence Exponent for the2D-Diffusion Equation and Related Kac Polynomials

Poplavskyi

Schehr

2018

Phys. Rev. Lett.

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We compute the persistence for the 2d-diffusion equation with random initial condition, i.e., the probability p0(t) that the diffusion field, at a given point x in the plane, has not changed sign up to time t. For large t, we show that p0(t) ∼ t −θ(2) with θ(2) = 3/16. Using the connection between the 2d-diffusion equation and Kac random polynomials, we show that the probability q0(n) that Kac polynomials, of (even) degree n, have no real root decays, for large n, as q0(n) ∼ n −3/4 . We obtain this result by using yet another connection with the truncated orthogonal ensemble of random matrices. This allows us to compute various properties of the zero-crossings of the diffusing field, equivalently of the real roots of Kac polynomials. Finally, we unveil a precise connection with a fourth model: the semi-infinite Ising spin chain with Glauber dynamics at zero temperature. arXiv:1806.11275v1 [cond-mat.stat-mech]

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What is the probability that a large random matrix has no real eigenvalues?

Abstract: We study the large-n limit of the probability p 2n,2k that a random 2nˆ2n matrix sampled from the real Ginibre ensemble has 2k real eigenvalues. We prove that, lim nÑ8 arXiv:1503.07926v1 [math.PR]

Cited by 19 publications

References 13 publications

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

How Many Eigenvalues of a Product of Truncated Orthogonal Matrices are Real?

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Exact Persistence Exponent for the2D-Diffusion Equation and Related Kac Polynomials

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