Painlevé II in Random Matrix Theory and Related Fields

Forrester, Peter J.; Witte, N. S.

doi:10.1007/s00365-014-9243-5

Cited by 22 publications

(55 citation statements)

References 65 publications

(72 reference statements)

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“…The pGUE arises naturally in the statistics of eigenvalues of GUE of random matrices. For

α \in N

and

ω = 1

, the pGUE can be interpreted as the probability density function of the classical GUE under the condition that μ is an eigenvalue with multiplicity α . Moreover, the distribution of the largest eigenvalue of this conditioning GUE can be expressed as the ratio of two Hankel determinants with different parameters defined in

Pro false(λ_{max} \leq μ false| λ = μ is an eigenvalue with multiplicity α false) = \frac{H_{n} false(μ; α, 0 false)}{H_{n} false(μ; α, 1 false)} .

Generally, if we remove each eigenvalue of the conditioning GUE with probability

ω \in [0, 1]

, then the distribution of the remaining largest eigenvalue is described as

Pro (λ_{max}^{R} \leq μ | λ = μ is an eigenvalue with multiplicity α) = \frac{H_{n} false(μ; α, ω false)}{H_{n} false(μ; α, 1 false)} .

The thinning and conditioning processes were introduced in random matrix theory in .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In , asymptotic formulas have been derived of the Hankel determinants associated with the Jacobi weight perturbed by a Fisher–Hartwig singularity close to the hard edge

x = 1

, involving the Jimbo–Miwa–Okamoto σ‐form of the Painlevé III equation. The Painlevé equations play an important role in the asymptotic study of the Hankel determinants; see , , , and .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Wu¹,

Zhao³

2017

Stud Appl Math

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We consider the Gaussian unitary ensemble perturbed by a Fisher–Hartwig singularity simultaneously of both root type and jump type. In the critical regime where the singularity approaches the soft edge, namely, the edge of the support of the equilibrium measure for the Gaussian weight, the asymptotics of the Hankel determinant and the recurrence coefficients, for the orthogonal polynomials associated with the perturbed Gaussian weight, are obtained and expressed in terms of a family of smooth solutions to the Painlevé XXXIV equation and the σ‐form of the Painlevé II equation. In addition, we further obtain the double scaling limit of the distribution of the largest eigenvalue in a thinning procedure of the conditioning Gaussian unitary ensemble, and the double scaling limit of the correlation kernel for the critical perturbed Gaussian unitary ensemble. The asymptotic properties of the Painlevé XXXIV functions and the σ‐form of the Painlevé II equation are also studied.

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“…The pGUE arises naturally in the statistics of eigenvalues of GUE of random matrices. For

α \in N

and

ω = 1

Pro false(λ_{max} \leq μ false| λ = μ is an eigenvalue with multiplicity α false) = \frac{H_{n} false(μ; α, 0 false)}{H_{n} false(μ; α, 1 false)} .

Generally, if we remove each eigenvalue of the conditioning GUE with probability

ω \in [0, 1]

, then the distribution of the remaining largest eigenvalue is described as

Pro (λ_{max}^{R} \leq μ | λ = μ is an eigenvalue with multiplicity α) = \frac{H_{n} false(μ; α, ω false)}{H_{n} false(μ; α, 1 false)} .

The thinning and conditioning processes were introduced in random matrix theory in .…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…In , asymptotic formulas have been derived of the Hankel determinants associated with the Jacobi weight perturbed by a Fisher–Hartwig singularity close to the hard edge

x = 1

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Wu¹,

Zhao³

2017

Stud Appl Math

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“…In the planar limit (large-N limit with µ 2 fixed), the theory has two phases -(I) SUSY phase for µ 2 > 2 and (II) SUSYbroken phase for µ 2 < 2. The phase (I) has infinitely degenerate minima parametrized have appeared repeatedly in the disguise of various combinatorial and statistical problems (see [15][16][17] for reviews), e.g. as a distribution of the length of the longest increasing subsequence in random permutations [18], as a distribution of particles in the asymmetric simple exclusion process [19,20], and as a one-dimensional surface growth process in the Karder-Parisi-Zhang universality class [21][22][23].…”

Section: )mentioning

confidence: 99%

“…(3.7) with the harmonic oscillator potential U (x) = N 2 x 2 , for which the orthogonal polynomials coincide with the Hermite polynomials: 16) and the orthonormal functions (3.11) become the wave functions of a particle under a onedimensional harmonic oscillator potential. In a simple large-N limit (planar limit), the eigenvalue density becomes…”

Section: Gue and Soft Edge Scaling Limitmentioning

confidence: 99%

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

Nishigaki

Sugino

2014

J. High Energ. Phys.

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We present an analytic expression of the nonperturbative free energy of a double-well supersymmetric matrix model in its double scaling limit, which corresponds to two-dimensional type IIA superstring theory on a nontrivial Ramond-Ramond background. To this end we draw upon the wisdom of random matrix theory developed by Tracy and Widom, which expresses the largest eigenvalue distribution of unitary ensembles in terms of a Painlevé II transcendent. Regularity of the result at any value of the string coupling constant shows that the third-order phase transition between a supersymmetry-preserving phase and a supersymmetry-broken phase, previously found at the planar level, becomes a smooth crossover in the double scaling limit. Accordingly, the supersymmetry is always broken spontaneously as its order parameter stays nonzero for the whole region of the coupling constant. Coincidence of the result with the unitary one-matrix model suggests that one-dimensional type 0 string theories partially correspond to the type IIA superstring theory. Our formulation naturally allows for introduction of an instanton chemical potential, and reveals the presence of a novel phase transition, possibly interpreted as condensation of instantons.

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“…We can view this as a gap probability in the soft edge GUE, conditioned to have an eigenvalue at −s. In such a setting, it is known [18,13] that the correlation kernel, to be denoted K soft s , can be written in terms of the usual soft edge scaled GUE kernel according to…”

Section: Conditioning With Fixed Eigenvaluesmentioning

confidence: 99%

Local Central Limit Theorem for Determinantal Point Processes

Forrester

Lebowitz

2014

J Stat Phys

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We prove a local central limit theorem (LCLT) for the number of points N (J) in a region J in R d specified by a determinantal point process with an Hermitian kernel. The only assumption is that the variance of N (J) tends to infinity as |J| → ∞. This extends a previous result giving a weaker central limit theorem (CLT) for these systems. Our result relies on the fact that the Lee-Yang zeros of the generating function for {E(k; J)} -the probabilities of there being exactly k points in Jall lie on the negative real z-axis. In particular, the result applies to the scaled bulk eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the Ginibre ensemble. For the GUE we can also treat the properly scaled edge eigenvalue distribution. Using identities between gap probabilities, the LCLT can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE). A LCLT is also established for the probability density function of the k-th largest eigenvalue at the soft edge, and of the spacing between k-th neigbors in the bulk.

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Painlevé II in Random Matrix Theory and Related Fields

Cited by 22 publications

References 65 publications

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Gaussian Unitary Ensemble with Boundary Spectrum Singularity and σ‐Form of the Painlevé II Equation

Tracy-Widom distribution as instanton sum of 2D IIA superstrings

Local Central Limit Theorem for Determinantal Point Processes

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