Universal Spectral Shocks in Random Matrix Theory --- Lessons for QCD

Blaizot, Jean‐Paul; Grela, Jacek; Nowak, Maciej A.; Warchoł, Piotr

doi:10.5506/aphyspolb.46.1785

Cited by 6 publications

(11 citation statements)

References 19 publications

(25 reference statements)

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“…(19), relegating the details to already published work [12,18]. First, it is exactly integrable on the complex plane for any initial conditions.…”

Section: Microscopic Limitmentioning

confidence: 99%

“…In order to simplify the analysis, let us start by performing a useful change of variables belonging to the class of Lamperti transformations [14][15][16][17]. This change of variables will allow to establish a connection between the Ornstein-Uhlenbeck process and the case of free diffusion (a = 0), where the results are known [12,18]. To check that indeed we can recover a free diffusion equation from the Ornstein-Uhlenbeck process, we explicitly write down the relevant Lamperti transformation:…”

Section: Microscopic Limitmentioning

confidence: 99%

“…It is worthy to disentangle the mixed variables present in the last, "drift" term, by repeating the Lamperti transformation defined by (18) in the hermitian case (with N replaced by 2N and Q instead of z). This change of variables leads to the free diffusion…”

Section: Diffusion In the Ginibre Ensemblementioning

confidence: 99%

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Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

et al. 2016

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Abstract. We compare the Ornstein-Uhlenbeck process for the Gaussian Unitary Ensemble to its non-hermitian counterpart -for the complex Ginibre ensemble. We exploit the mathematical framework based on the generalized Green's functions, which involves a new, hidden complex variable, in comparison to the standard treatment of the resolvents. This new variable turns out to be crucial to understand the pattern of the evolution of non-hermitian systems. The new feature is the emergence of the coupling between the flow of eigenvalues and that of left/right eigenvectors. We analyze local and global equilibria for both systems. Finally, we highlight some unexpected links between both ensembles.

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“…(19), relegating the details to already published work [12,18]. First, it is exactly integrable on the complex plane for any initial conditions.…”

Section: Microscopic Limitmentioning

confidence: 99%

Section: Microscopic Limitmentioning

confidence: 99%

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Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

et al. 2016

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“…Let us mention that for ν = − (24) and is called the symmetric Pearcey integral through its connection with the symmetric Pearcey kernel arising for phenomena of random surface growth with a wall [40]. Moreover, for positive integer ν, as the Wishart ensemble is connected to Chiral random matrices, it has an analog in the integral describing the statistical properties of the Dirac operator around its zero eigenvalue, at the moment of chiral symmetry breaking in Euclidean Quantum Chromodynamics [46]. The averaged characteristic polynomial of a diffusing complex chiral matrix is namely defined bỹ…”

Section: The Characteristic Polynomial At the Critical Pointmentioning

confidence: 99%

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

2014

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We study the diffusion of complex Wishart matrices and derive a partial differential equation governing the behavior of the associated averaged characteristic polynomial. In the limit of large size matrices, the inverse Cole-Hopf transform of this polynomial obeys a nonlinear partial differential equation whose solutions exhibit shocks at the evolving edges of the eigenvalue spectrum. In a particular scenario one of those shocks hits the origin that plays the role of an impassable wall. To investigate the universal behavior in the vicinity of this wall, a critical point, we derive an integral representation for the averaged characteristic polynomial and study its asymptotic behavior. The result is a Bessoid function.

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“…and is called the symmetric Pearcey integral through its connection with the symmetric Pearcey kernel arising for phenomena of random surface growth with a wall [40]. Moreover, for positive integer ν, as the Wishart ensemble is connected to Chiral random matrices, it has an analog in the integral describing the statistical properties of the Dirac operator around its zero eigenvalue, at the moment of chiral symmetry breaking in Euclidean Quantum Chromodynamics [46]. The averaged characteristic polynomial of a diffusing complex chiral matrix is namely defined by…”

Section: The Characteristic Polynomial At the Critical Pointmentioning

confidence: 99%

Universal shocks in the Wishart random-matrix ensemble

2013

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We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple inviscid Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution, through the method of characteristics, presents singularities that we relate to the precursors of shock formation in the Burgers equation. The finite N effects appear as a viscosity term in the Burgers equation. Using a scaling analysis of the complete equation for the characteristic polynomial, in the vicinity of the shocks, we recover in a simple way the universal Bessel oscillations (so-called hard-edge singularities) familiar in random-matrix theory.

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Universal Spectral Shocks in Random Matrix Theory --- Lessons for QCD

Cited by 6 publications

References 19 publications

Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

Ornstein–Uhlenbeck diffusion of hermitian and non-hermitian matrices—unexpected links

Universal shocks in the Wishart random-matrix ensemble. II. Nontrivial initial conditions

Universal shocks in the Wishart random-matrix ensemble

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