Biorthogonal ensembles

Borodin, Alexei

doi:10.1016/s0550-3213(98)00642-7

Cited by 221 publications

(382 citation statements)

References 20 publications

(5 reference statements)

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“…, x n ) = 1 Z n det (f i (x j )) n i,j=1 det (g i (x j )) n i,j=1 and with Z n a normalization constant. Note that (2.3) is a biorthogonal ensemble [10]. In particular, it is a determinantal point process with correlation kernel K n (x, y) = n i,j=1 .…”

Section: Correlation Kernel and The Riemann-hilbert Problemmentioning

confidence: 99%

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Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

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We study a model of n one-dimensional non-intersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time-space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new familiy of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new RiemannHilbert problem of size 4 × 4. We prove solvability of the Riemann-Hilbert problem and establish a remarkable connection with the Hastings-McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the non-intersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class.

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Section: Correlation Kernel and The Riemann-hilbert Problemmentioning

confidence: 99%

“…9) 10) and consider the following RH problem which was introduced in [14] as a generalization of the RH problem for orthogonal polynomials [24], see also [45].…”

Section: Correlation Kernel and The Riemann-hilbert Problemmentioning

confidence: 99%

Critical behavior of nonintersecting Brownian motions at a tacnode

Delvaux

Kuijlaars

Zhang

2011

Comm Pure Appl Math

129

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“…The classical cases (Hermite, Laguerre and Jacobi) were worked out in [24]. Note that (2.2) is exactly the type of ensemble that (2.1) leads us to consider since:…”

Section: Biorthogonal Stieltjes-wigertmentioning

confidence: 99%

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Dolivet

Tierz

2007

Journal of Mathematical Physics

108

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Abstract. Employing the random matrix formulation of Chern-Simons theory on Seifert manifolds, we show how the Stieltjes-Wigert orthogonal polynomials are useful in exact computations in Chern-Simons matrix models. We construct a biorthogonal extension of the Stieltjes-Wigert polynomials, not available in the literature, necessary to study Chern-Simons matrix models when the geometry is a lens space. We also discuss several other results based on the properties of the polynomials: the equivalence between the StieltjesWigert matrix model and the discrete model that appears in q-2D Yang-Mills and the relationship with Rogers-Szegö polynomials and the corresponding equivalence with an unitary matrix model. Finally, we also give a detailed proof of a result that relates quantum dimensions with averages of Schur polynomials in the Stieltjes-Wigert ensemble.

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“…Hence, (1.5) is a biorthogonal ensemble [9], which is a special case of a determinantal point process. The correlation kernel is given by K n (x, y) = n j=1 φ j (x)ψ j (y), (1.7) where the functions φ j , ψ j , j = 1, .…”

Section: Introductionmentioning

confidence: 99%

“…To obtain interesting results, we make a time scaling as in [38] so that the time variable depends on the number of paths n. That is, we replace the variables t and T by 9) so that 0 < t < 1. Now, letting n → ∞, the paths fill out a region in the tx-plane that looks like one of the regions shown in Figures 1 and 2, depending on the product ab.…”

Section: Introductionmentioning

confidence: 99%

Non-intersecting squared Bessel paths with one positive starting and ending point

Delvaux

Kuijlaars

Román

et al. 2012

JAMA

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We consider a model of n non-intersecting squared Bessel processes with one starting point a > 0 at time t = 0 and one ending point b > 0 at time t = T . After proper scaling, the paths fill out a region in the tx-plane. Depending on the value of the product ab the region may come to the hard edge at 0, or not. We formulate a vector equilibrium problem for this model, which is defined for three measures, with upper constraints on the first and third measures and an external field on the second measure. It is shown that the limiting mean distribution of the paths at time t is given by the second component of the vector that minimizes this vector equilibrium problem. The proof is based on a steepest descent analysis for a 4 × 4 matrix valued Riemann-Hilbert problem which characterizes the correlation kernel of the paths at time t. We also discuss the precise locations of the phase transitions.

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Biorthogonal ensembles

Cited by 221 publications

References 20 publications

Critical behavior of nonintersecting Brownian motions at a tacnode

Critical behavior of nonintersecting Brownian motions at a tacnode

Chern-Simons matrix models and Stieltjes-Wigert polynomials

Non-intersecting squared Bessel paths with one positive starting and ending point

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