Large deviations for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distribution

Butez, Raphaël

doi:10.1214/17-ecp68

Cited by 10 publications

(13 citation statements)

References 20 publications

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“…In the large n limit the particles have an almost sure limiting empirical measure µ * which minimizes a corresponding equilibrium problem [9,11,21]. The equilibrium problem was studied in detail in [13].…”

Section: The Muttalib-borodin Ensemblementioning

confidence: 99%

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Kuijlaars

Leslie

2019

Nonlinearity

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The Muttalib-Borodin biorthogonal ensemble is a probability density function for n particles on the positive real line that depends on a parameter θ and an external field V . For θ = 1 2 we find the large n behavior of the associated correlation kernel with only few restrictions on V . The idea is to relate the ensemble to a type II multiple orthogonal polynomial ensemble that can in turn be related to a 3 × 3 Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest descent method. The main ingredient is the construction of the local parametrix at the origin, with the help of Meijer G-functions, and its matching condition with a global parametrix. We will present a new iterative technique to obtain the matching condition, which we expect to be applicable in more general situations as well.with limiting kernel.is Wright's generalization of the Bessel function. For θ = 1, the limit (1.5) reduces to the well-known Bessel kernel in the theory of random matrices. Several other expressions are known for the kernel (1.6). For example, Zhang [40, Theorem 1.2] gives a double contour integral, and if θ or 1/θ is an integer, then (1.5) can be expressed in terms of Meijer G-functions [30]. These so-called Meijer G-kernels appear in singular value distributions for products of random matrices [2, 3] and in their hard edge scaling limit [4,5,23,24,25,31]. See [1] for a survey paper. The bulk and soft edge scaling limits for singular values of Ginibre random matrices are the usual sine and Airy kernels [32], and these classical limits were also established for the Muttalib-Borodin model in the Laguerre case [40].It is natural to expect that the limit (1.5) is not restricted to the case V (x) = x, but holds for much more general external fields. In this paper we consider θ = 1 2 and we show that the hard edge scaling limit (1.5) indeed holds for a large class of external fields V .

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Section: The Muttalib-borodin Ensemblementioning

confidence: 99%

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

Kuijlaars

Leslie

2019

Nonlinearity

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“…In the large n limit, the particles have an almost sure limiting measure µ * which is the minimizer of among all probability measures µ on [0, ∞). This follows from large deviation results for (1.1) and related models that were studied in [3,5,10]. For θ = 1, the functional (1.2) reduces to the usual energy in the presence of an external field [20].…”

Section: The Muttalib-borodin Ensemblementioning

confidence: 82%

A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

Kuijlaars

2016

SIGMA

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“…On the other hand, near a soft edge, this density vanishes to the order 1/2 for any value of θ ; this is the usual square root behavior that is often encountered in random matrix theory. We also refer to [5,8,23] for related results on the equilibrium measure.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…3.1 and 3.2 , respectively. We let {c j = c j (θ, α)} 8 1 and {b j = b j (θ, α)} 2 1 be the constants defined in (2.4) and (2.7), respectively. Proposition 3.1 (Asymptotics of ln G(ζ )).…”

Section: Asymptotics Of G(ζ ) and P(ζ )mentioning

confidence: 99%

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

Charlier

Lenells

Mauersberger

2021

Commun. Math. Phys.

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We consider the limiting process that arises at the hard edge of Muttalib–Borodin ensembles. This point process depends on $$\theta > 0$$ θ > 0 and has a kernel built out of Wright’s generalized Bessel functions. In a recent paper, Claeys, Girotti and Stivigny have established first and second order asymptotics for large gap probabilities in these ensembles. These asymptotics take the form $$\begin{aligned} {\mathbb {P}}(\text{ gap } \text{ on } [0,s]) = C \exp \left( -a s^{2\rho } + b s^{\rho } + c \ln s \right) (1 + o(1)) \qquad \text{ as } s \rightarrow + \infty , \end{aligned}$$ P ( gap on [ 0 , s ] ) = C exp - a s 2 ρ + b s ρ + c ln s ( 1 + o ( 1 ) ) as s → + ∞ , where the constants $$\rho $$ ρ , a, and b have been derived explicitly via a differential identity in s and the analysis of a Riemann–Hilbert problem. Their method can be used to evaluate c (with more efforts), but does not allow for the evaluation of C. In this work, we obtain expressions for the constants c and C by employing a differential identity in $$\theta $$ θ . When $$\theta $$ θ is rational, we find that C can be expressed in terms of Barnes’ G-function. We also show that the asymptotic formula can be extended to all orders in s.

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Large deviations for biorthogonal ensembles and variational formulation for the Dykema-Haagerup distribution

Cited by 10 publications

References 20 publications

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

The local universality of Muttalib–Borodin biorthogonal ensembles with parameter $\theta = \frac{1}{2}$

A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

Higher Order Large Gap Asymptotics at the Hard Edge for Muttalib–Borodin Ensembles

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