Random Matrices with Equispaced External Source

Claeys, Tom; Wang, Dong

doi:10.1007/s00220-014-1988-y

Cited by 32 publications

(60 citation statements)

References 36 publications

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“…2 periodic functions e πω 1 y , e πω 2 y and with respect to the weight e −W(y) supported on R. As shown by Claeys and Wang [46] for a specific degeneration (which corresponds basically to sending one of the ω's in (1.5.27) to zero) and then in full extent by Claeys and Romano [45], such a system of bi-orthogonal polynomials solves a vector Riemann-Hilbert problem. Furthermore, the non-linear steepest descent approach [54,55] to the uniform in the variable large degree-N asymptotics of orthogonal polynomials can be generalised to such a bi-orthogonal setting, leading to Plancherel-Rotach like asymptotics for these bi-orthogonal polynomials [46].…”

Section: Putting In Perspective the Bi-orthogonal Ensemblesmentioning

confidence: 89%

“…The case f (λ) = λ θ and g(λ) = λ is of special interest, since the bi-orthogonal polynomials can be effectively described. In [21] Borodin was able to establish certain universality results for specific examples of confining potentials V. Furthermore, it was observed, first on a specific example by Claeys and Wang [46] and then in full generality by Claeys and Romano [45] that the bi-orthogonal polynomials can be characterised by means of a Riemann-Hilbert problem. However, for the moment, the Riemann-Hilbert problem-based machinery still did not lead to the asymptotic evaluation of the associated partition functions.…”

Section: Biorthogonal Ensemblesmentioning

confidence: 94%

“…Furthermore, the non-linear steepest descent approach [54,55] to the uniform in the variable large degree-N asymptotics of orthogonal polynomials can be generalised to such a bi-orthogonal setting, leading to Plancherel-Rotach like asymptotics for these bi-orthogonal polynomials [46]. In principle, by adapting the steps of [66], one should be able to derive the large-N asymptotic expansion of the integral z N [W] in presence of varying weights, viz.…”

Section: Putting In Perspective the Bi-orthogonal Ensemblesmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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This paper develops a method to carry out the large-N asymptotic analysis of a class of Ndimensional integrals arising in the context of the so-called quantum separation of variables method. We push further ideas developed in the context of random matrices of size N, but in the present problem, two scales 1/N α and 1/N naturally occur. In our case, the equilibrium measure is N α -dependent and characterised by means of the solution to a 2×2 Riemann-Hilbert problem, whose large-N behaviour is analysed in detail. Combining these results with techniques of concentration of measures and an asymptotic analysis of the Schwinger-Dyson equations at the distributional level, we obtain the large-N behaviour of the free energy explicitly up to o(1). The use of distributional Schwinger-Dyson is a novelty that allows us treating sufficiently differentiable interactions and the mixing of scales 1/N α and 1/N, thus waiving the analyticity assumptions often used in random matrix theory.1 gborot@mpim-bonn.mpg.de 2 guionnet@math.mit.edu 3 karol.kozlowski@ens-lyon.fr 2 An opening discussionThe present work develops techniques enabling one to carry out the large-N asymptotic analysis of a class of multiple integrals that arise as representations for the correlation functions in quantum integrable models solvable by the quantum separation of variables. We shall refer to the general class of such integrals as the sinh model: [9,59,60,82,106,105,139,146]. The expressions obtained there are either directly of the form given above or are amenable to this form (with, possibly, a change of the integration contour from R N to C N , with C a curve in C) upon elementary manipulations. Furthermore, a degeneration of z N [W] arises as a multiple integral representation for the partition function of the six-vertex model subject to domain wall boundary conditions [93]. In the context of quantum integrable systems, the number N of integrals defining z N is related to the number of sites in a model (as, e.g. in the case of the compact or non-compact XXZ chains or the lattice regularisations of the Sinh or Sine-Gordon models) or to the number of particles (as, e.g. in the case of the quantum Toda chain). From the point of view of applications, one is mainly interested in the thermodynamic limit of the model, which is attained by sending N to +∞. For instance, in the case of an integrable lattice discretisation of some quantum field theory, one obtains in this way an exact and non-perturbative description of a quantum field theory in 1 + 1 dimensions and in finite volume. This limit, at the level of z N [W], translates itself in the need to extract the large N-asymptotic expansion of ln z N [W] up to o(1). It is, in fact, the constant term in the expansion of ln zwith W ′ some deformation of W that gives rise to the correlation functions of the underlying quantum field theory in finite volume. These applications to physics constitute the first motivation for our analysis. From the purely mathematical side, the motivation of our works stems ...

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Section: Putting In Perspective the Bi-orthogonal Ensemblesmentioning

confidence: 89%

Section: Biorthogonal Ensemblesmentioning

confidence: 94%

Section: Putting In Perspective the Bi-orthogonal Ensemblesmentioning

confidence: 99%

See 1 more Smart Citation

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Borot¹,

Guionnet²,

Kozłowski³

2016

Mathematical Physics Studies

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“…The matrix D n is usually referred to as an external source in the mathematical physics literature (see [3,7,8,10] and references therein).…”

Section: 2mentioning

confidence: 99%

“…In [10], Claeys and Wang study an ensemble of random matrices that generalizes (9) when D n has equispaced eigenvalues.…”

Section: 2mentioning

confidence: 99%

Universality of local eigenvalue statistics in random matrices with external source

O’Rourke

2014

Random Matrices: Theory Appl.

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Abstract. Consider a random matrix of the form Wn := 1 √ n Mn + Dn, where Mn is a Wigner matrix and Dn is a real deterministic diagonal matrix (Dn is commonly referred to as an external source in the mathematical physics literature). We study the universality of the local eigenvalue statistics of Wn for a general class of Wigner matrices Mn and diagonal matrices Dn.Unlike the setting of many recent results concerning universality, the global semicircle law fails for this model. However, we can still obtain the universal sine kernel formula for the correlation functions. This demonstrates the remarkable phenomenon that local laws are more resilient than global ones.The universality of the correlation functions follows from a four moment theorem, which we prove using a variant of the approach used earlier by Tao and Vu.

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Asymptotics of Muttalib–Borodin determinants with Fisher–Hartwig singularities

Charlier

2022

Sel. Math. New Ser.

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Random Matrices with Equispaced External Source

Cited by 32 publications

References 36 publications

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Asymptotic Expansion of a Partition Function Related to the Sinh-model

Universality of local eigenvalue statistics in random matrices with external source

Asymptotics of Muttalib–Borodin determinants with Fisher–Hartwig singularities

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