Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

Kuijlaars, Arno B. J.; McLaughlin, Kenneth D.T-R

doi:10.1002/(sici)1097-0312(200006)53:6<736::aid-cpa2>3.0.co;2-5

Cited by 112 publications

(156 citation statements)

References 36 publications

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“…We require V to be such that the following generic conditions hold: the density ψ V is strictly positive on (a, b), it vanishes like a square root at the endpoints a, b (in other words, h(x) > 0 for x ∈ [a, b]), and the variational inequality (1.8) is strict on R\ [a, b]. If all those assumptions hold, we say that V is one-cut regular (see [26] for a general classification of singularities). In addition, we require zero to be contained in the interior of the support of µ V , i.e., a < 0 < b.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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Abstract. We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the formwhere M is an n × n Hermitian matrix, α > −1/2 and t ∈ R, in double scaling limits where n → ∞ and simultaneously t → 0. If t is proportional to 1/n 2 , a transition takes place which can be described in terms of a family of solutions to the Painlevé V equation. These Painlevé solutions are in general transcendental functions, but for certain values of α, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel.

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Section: Statement Of Resultsmentioning

confidence: 99%

“…As n → ∞ and simultaneously t → 0 in such a way that τ n,t → τ = 0, there exists a limiting kernel K PV α such that 26) for any u, v ∈ R if τ < 0, and for any u, v ∈ R\{± √ τ /4} if τ > 0.…”

mentioning

confidence: 99%

Random Matrices with Merging Singularities and the Painlevé V Equation

Claeys

Fahs²

2016

SIGMA

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“…For unitarily invariant random matrix ensembles (1.16) the local correlations near edge points are expressed in the limit n → ∞ in terms of the Airy kernel 21) provided that the limiting mean eigenvalue density vanishes like a square root, which is generically the case [19]. In our non-unitarily invariant random matrix model, the limiting mean eigenvalue density vanishes like a square root, (1.12), and indeed we recover the kernel (1.21) in the limit n → ∞.…”

Section: Remarkmentioning

confidence: 99%

“…Consider the disk D(z 1 , r). In the regions I and IV, we have by (7.7), (8.1) and (8.6) 19) hence by (4.1), (7.14), and (7.15) By restricting equation (10.20) to the (1, 1) entry, and using the first expression of (7.13) (in region I) or the fourth expression of (7.13) (in region IV) to evaluate Φ(n 2/3 β(z)), and (6.16) to evaluate M(z), we obtain that P n (z)e − n 2 z 2 = √ π n 1/6 B(z)Ai(n 2/3 β(z))(1 + O(n −1 )) +n −1/6 C(z)Ai ′ (n 2/3 β(z))(1 + O(n −1 )) e −nα(z)+nl 1 , The same asymptotics, (10.22), holds in regions II and III as well. Thus, (10.22) holds in the full disk D(z 1 , r).…”

Section: Large N Asymptotics Of the Multiple Hermite Polynomialsmentioning

confidence: 99%

Large n Limit of Gaussian Random Matrices with External Source, Part I

Bleher

Kuijlaars

2004

Commun. Math. Phys.

132

271

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Abstract. We consider the random matrix ensemble with an external source 1defined on n× n Hermitian matrices, where A is a diagonal matrix with only two eigenvalues ±a of equal multiplicity. For the case a > 1, we establish the universal behavior of local eigenvalue correlations in the limit n → ∞, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a 3 × 3-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large n limit.

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“…If one of these conditions is not valid, V s,t is called singular. The singular points x * are classified as follows, see [10,21]: (i) x * ∈ R \ S s,t is a type I singular point if equality in (1.9) holds. Then, x * is a zero of q + s,t of multiplicity 4m with m ∈ N.…”

Section: Unitary Random Matrix Ensemblesmentioning

confidence: 99%

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

Claeys

Vanlessen

2007

Commun. Math. Phys.

115

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We consider unitary random matrix ensembles Z −1 n,s,t e −n tr Vs,t(M) dM on the space of Hermitian n × n matrices M , where the confining potential V s,t is such that the limiting mean density of eigenvalues (as n → ∞ and s, t → 0) vanishes like a power 5/2 at a (singular) endpoint of its support. The main purpose of this paper is to prove universality of the eigenvalue correlation kernel in a double scaling limit. The limiting kernel is built out of functions associated with a special solution of the P 2 I equation, which is a fourth order analogue of the Painlevé I equation. In order to prove our result, we use the wellknown connection between the eigenvalue correlation kernel and the Riemann-Hilbert (RH) problem for orthogonal polynomials, together with the Deift/Zhou steepest descent method to analyze the RH problem asymptotically. The key step in the asymptotic analysis will be the construction of a parametrix near the singular endpoint, for which we use the model RH problem for the special solution of the P 2 I equation. In addition, the RH method allows us to determine the asymptotics (in a double scaling limit) of the recurrence coefficients of the orthogonal polynomials with respect to the varying weights e −nVs,t on R. The special solution of the P 2 I equation pops up in the n −2/7 -term of the asymptotics.

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Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

Cited by 112 publications

References 36 publications

Random Matrices with Merging Singularities and the Painlevé V Equation

Random Matrices with Merging Singularities and the Painlevé V Equation

Large n Limit of Gaussian Random Matrices with External Source, Part I

Universality of a Double Scaling Limit near Singular Edge Points in Random Matrix Models

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