Generalized random matrix model with additional interactions

Yadav, Swapnil; Alam, Kazi; Muttalib, K. A.; Wang, Dong

doi:10.1088/1751-8121/ab56e0

Cited by 6 publications

(14 citation statements)

References 18 publications

(31 reference statements)

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“…Densities for γ-Biorthogonal ensemble are shown in figure 7. Effective potentials for γ = 0.4, 0.8 and θ = 2 were used from [YAMW19]. Potential was taken to be V (x) = 2x.…”

Section: Verification Of Known Resultsmentioning

confidence: 99%

“…In particular, for Muttalib-Borodin ensembles with r(x) = x, s(x) = x θ , it allows us to compute the kernel for arbitrary θ and arbitrary confining potential. Moreover, the generalized MB ensemble with an additional parameter γ, called the γ-ensembles [YAMW19], are shown to be equivalent to the MB ensembles with an effective γ-dependent potential. Given this effective potential, it should in principle be possible to use the present method to obtain the kernel for the γ-ensembles as well.…”

Section: Ensembles Without the Polynomialsmentioning

confidence: 99%

“…e −Vc(x) is the Askey weight for unitary critical ensemble [MCIN93], see (4). See [YAMW19] for γ-ensembles. Of these, the statistical properties of the first two (Wigner-Dyson and critical) are well-known, and we will reproduce them to show the validity of our method.…”

Section: Statistical Analysis Of Level Sequencementioning

confidence: 99%

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On the computation of density and two-point correlation functions of a class of random matrix ensembles

Alam,

Yadav,

Muttalib

2020

Preprint

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We demonstrate a method to solve a general class of random matrix ensembles numerically. The method is suitable for solving log-gas models with biorthogonal type two-body interactions and arbitrary potentials. We reproduce standard results for a variety of well-known ensembles and show some new results for the Muttalib-Borodin ensembles and recently introduced γ-ensemble for which analytic resultsare not yet available.

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“…Densities for γ-Biorthogonal ensemble are shown in figure 7. Effective potentials for γ = 0.4, 0.8 and θ = 2 were used from [YAMW19]. Potential was taken to be V (x) = 2x.…”

Section: Verification Of Known Resultsmentioning

confidence: 99%

Section: Ensembles Without the Polynomialsmentioning

confidence: 99%

See 1 more Smart Citation

On the computation of density and two-point correlation functions of a class of random matrix ensembles

Alam,

Yadav,

Muttalib

2020

Preprint

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“…The corresponding gap probabilities near the origin are investigated in [17,18,20,60] and the local limits of K n away from the origin for the classical weights are given in [33] and [59]. The limiting mean distribution of the particles in Muttalib-Borodin ensemble is formulated as the minimizer of a (vector) equilibrium problem in [21,38], and the large deviation results can be found in [13,15,23,28]; see also [16,42,58] for other investigations and extensions of the Muttalib-Borodin ensemble.…”

Section: The Modelmentioning

confidence: 99%

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

Wang¹,

Zhang²

2021

Preprint

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In this paper, we consider Muttalib-Borodin ensemble of Laguerre type, a determinantal point process over [0, ∞) which depends on the varying weights x α e −nV (x) , α > −1, and a parameter θ. For θ being a positive integer, we derive asymptotics of the associated biorthogonal polynomials near the origin for a large class of potential functions V as n → ∞. This further allows us to establish the hard edge scaling limit of the correlation kernel, which is previously only known in the special cases and conjectured to be universal. Our proof is based on the Deift/Zhou nonlinear steepest descent analysis of two 1 × 2 vector-valued Riemann-Hilbert problems that characterize the biorthogonal polynomials and the explicit constructions of (θ + 1) × (θ + 1) local parametrices near the origin in terms of the Meijer G-functions.

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“…A generalized random matrix model with additional interactions [1], called the γ-ensembles, was introduced recently as a solvable toy model for three-dimensional (3D) disordered conductors. The joint probability distribution (jpd) of the N non-negative eigenvalues x i for these γ-ensembles has the form…”

Section: Introductionmentioning

confidence: 99%

Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model

et al. 2021

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We consider a limiting case of the joint probability distribution for a random matrix ensemble with an additional interaction term controlled by an exponent γ (called the γ-ensembles), which is equivalent to the probability distribution of Laguerre β-ensembles. The effective potential, which is essentially the confining potential for an equivalent ensemble with γ = 1 (called the Muttalib-Borodin ensemble), is a crucial quantity defined in solution to the Riemann-Hilbert problem associated with the γ-ensembles. It enables us to numerically compute the eigenvalue density of Laguerre β-ensembles for all β > 1. In addition, the effective potential shows a non-monotonic behavior for γ < 1, suggestive of its possible role in developing a transition from a diverging to a non-diverging density near the hard-edge of the eigenvalue spectrum. Such a transition in eigenvalue density is of critical importance for the random matrix theory describing disordered mesoscopic systems. For γ-ensembles with γ > 1, the effective potential also shows transition from non-monotonic to monotonic behavior near the origin when a parameter of the additional interaction term is varied.

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Generalized random matrix model with additional interactions

Cited by 6 publications

References 18 publications

On the computation of density and two-point correlation functions of a class of random matrix ensembles

On the computation of density and two-point correlation functions of a class of random matrix ensembles

A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles

Nonmonotonic confining potential and eigenvalue density transition for generalized random matrix model

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