| 1. |
- Abathun, Addisalem, 1978-
(författare)
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Asymptotic distribution of zeros of a certain class of hypergeometric polynomials
- 2014
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The thesis consists of two papers, both treating hypergeometric polynomials, and a short introduction. The main results are as follows.In the first paper,we study the asymptotic zero distribution of a family of hypergeometric polynomials in one complex variable as their degree goes to infinity,using the associated differential equations that hypergeometric polynomials satisfy. We describe in particular the curve complex on which the zeros cluster, as level curves associated to integrals on an algebraic curve derived from the equation. The new result is first of all that we are able to formulate results on the location of zeros of generalized hypergeometric polynomials in greater generality than before (earlier results are mainly concerned with the Gauss hypergeometric case.) Secondly, we are able to formulate a precise conjucture giving the asymptotic behaviour of zeros in the generalized case of our polynomials, which covers previous results.In the second paper we partly prove one of the conjectures in the first paper by using Euler integral representation of the Gauss hypergeometric functions together with the Saddle point method.
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| 2. |
- Akemann, G., et al.
(författare)
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The elliptic Ginibre ensemble : A unifying approach to local and global statistics for higher dimensions
- 2023
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 64:2
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Tidskriftsartikel (refereegranskat)abstract
- The elliptic Ginibre ensemble of complex non-Hermitian random matrices allows us to interpolate between the rotationally invariant Ginibre ensemble and the Gaussian unitary ensemble of Hermitian random matrices. It corresponds to a two-dimensional one-component Coulomb gas in a quadrupolar field at inverse temperature beta = 2. Furthermore, it represents a determinantal point process in the complex plane with the corresponding kernel of planar Hermite polynomials. Our main tool is a saddle point analysis of a single contour integral representation of this kernel. We provide a unifying approach to rigorously derive several known and new results of local and global spectral statistics, including in higher dimensions. First, we prove the global statistics in the elliptic Ginibre ensemble first derived by Forrester and Jancovici [Int. J. Mod. Phys. A 11, 941 (1996)]. The limiting kernel receives its main contribution from the boundary of the limiting elliptic droplet of support. In the Hermitian limit, there is a known correspondence between non-interacting fermions in a trap in d real dimensions R-d and the d-dimensional harmonic oscillator. We present a rigorous proof for the local d-dimensional bulk (sine) and edge (Airy) kernel first defined by Dean et al. [Europhys. Lett. 112, 60001 (2015)], complementing the recent results by Deleporte and Lambert [arXiv:2109.02121 (2021)]. Using the same relation to the d-dimensional harmonic oscillator in d complex dimensions C-d, we provide new local bulk and edge statistics at weak and strong non-Hermiticity, where the former interpolates between correlations in d real and d complex dimensions. For C-d with d = 1, this corresponds to non-interacting fermions in a rotating trap.
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| 3. |
- Berggren, Tomas, et al.
(författare)
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Correlation functions for determinantal processes defined by infinite block Toeplitz minors
- 2019
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Ingår i: Advances in Mathematics. - : ACADEMIC PRESS INC ELSEVIER SCIENCE. - 0001-8708 .- 1090-2082. ; 356
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Tidskriftsartikel (refereegranskat)abstract
- We study the correlation functions for determinantal point processes defined by products of infinite minors of block Toeplitz matrices. The motivation for studying such processes comes from doubly periodically weighted tilings of planar domains, such as the two-periodic Aztec diamond. Our main results are double integral formulas for the correlation kernels. In general, the integrand is a matrix-valued function built out of a factorization of the matrix-valued weight. In concrete examples the factorization can be worked out in detail and we obtain explicit integrands. In particular, we find an alternative proof for a formula for the two-periodic Aztec diamond recently derived in [20]. We strongly believe that also in other concrete cases the double integral formulas are good starting points for asymptotic studies.
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| 4. |
- Berggren, Tomas, et al.
(författare)
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Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble
- 2017
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Ingår i: Mathematical physics, analysis and geometry. - : SPRINGER. - 1385-0172 .- 1572-9656. ; 20:3
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the asymptotic behavior of mesoscopic fluctuations for the thinned Circular Unitary Ensemble. The effect of thinning is that the eigenvalues start to decorrelate. The decorrelation is stronger on the larger scales than on the smaller scales. We investigate this behavior by studying mesoscopic linear statistics. There are two regimes depending on the scale parameter and the thinning parameter. In one regime we obtain a CLT of a classical type and in the other regime we retrieve the CLT for CUE. The two regimes are separated by a critical line. On the critical line the limiting fluctuations are no longer Gaussian, but described by infinitely divisible laws. We argue that this transition phenomenon is universal by showing that the same transition and their laws appear for fluctuations of the thinned sine process in a growing box. The proofs are based on a Riemann-Hilbert problem for integrable operators.
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| 5. |
- Berggren, Tomas
(författare)
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On determinantal point processes and random tilings with doubly periodic weights
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis is dedicated to asymptotic analysis of determinantal point processes originating from random matrix theory and random tiling models. Our main interest lies in random tilings of planar domains with doubly periodic weights.Uniformly distributed random tiling models are known to be a very rich class of models where many interesting phenomena can be observed. These models have therefore been under investigation for many years and many aspects of the models are by now well understood. Random tiling models with doubly periodic weights are in fact an even richer class of models. However, these models are much more difficult to analyze and for a thorough study of their behavior new ideas are needed. This thesis increases the understanding of random tiling models with doubly periodic weights.The thesis consists of three papers and two chapters; one introductory and background chapter and one chapter giving an overview of the papers.Paper A deals with linear statistics of the thinned Circular Unitary Ensemble and the thinned sine process. The thinning creates a transition from the Circular Unitary Ensemble respectively sine process to the Poisson process. We study a part of these transitions in detail.In Papers B and C we study random tiling models with doubly periodic weights. These two papers constitute the main contribution of this thesis.In Paper B we give a general method how to analyze a large family of random tiling models. In particular, we provide a double integral formula for the correlation kernel in terms of a Wiener-Hopf factorization of an associated matrix-valued function. We also present a recursive method on how to construct the Wiener-Hopf factorization.The method developed in Paper B is used in Paper C to analyze the 2×k-periodic Aztec diamond. More precisely, we derive the correlation kernel for the Aztec diamond of finite size and give a detailed description of the model as the size tends to infinity.
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| 6. |
- Borodin, Alexei, et al.
(författare)
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Biased 2 × 2 periodic Aztec diamond and an elliptic curve
- 2023
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Ingår i: Probability theory and related fields. - : Springer Nature. - 0178-8051 .- 1432-2064. ; 187:1-2, s. 259-315
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Tidskriftsartikel (refereegranskat)abstract
- We study random domino tilings of the Aztec diamond with a biased 2 × 2 periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight.
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| 7. |
- Borodin, Alexei, et al.
(författare)
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Limits of determinantal processes near a tacnode
- 2011
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Ingår i: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. - 0246-0203. ; 47:1, s. 243-258
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Tidskriftsartikel (refereegranskat)abstract
- We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter epsilon > 0. The domain has two cusps, one pointing up and one pointing down. In the limit epsilon down arrow 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime epsilon down arrow 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process.
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| 8. |
- Breuer, Jonathan, et al.
(författare)
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Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients
- 2017
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Ingår i: Journal of The American Mathematical Society. - : American Mathematical Society (AMS). - 0894-0347 .- 1088-6834. ; 30:1, s. 27-66
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Tidskriftsartikel (refereegranskat)abstract
- We study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles. We study those biorthogonal ensembles for which the underlying biorthogonal family satisfies a finite term recurrence and describe the asymptotic fluctuations using right limits of the recurrence matrix. As a consequence, we show that whenever the right limit is a Laurent matrix, a central limit theorem holds. We will also discuss the implications for orthogonal polynomial ensembles. In particular, we obtain a central limit theorem for the orthogonal polynomial ensemble associated with any measure belonging to the Nevai class of an interval. Our results also extend previous results on unitary ensembles in the one-cut case. Finally, we will illustrate our results by deriving central limit theorems for the Hahn ensemble for lozenge tilings of a hexagon and for the Hermitian two matrix model.
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| 9. |
- Breuer, Jonathan, et al.
(författare)
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Nonintersecting paths with a staircase initial condition
- 2012
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Ingår i: Electronic Journal of Probability. - 1083-6489. ; 17, s. 1-24
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Tidskriftsartikel (refereegranskat)abstract
- We consider an ensemble of N discrete nonintersecting paths starting from equidistant points and ending at consecutive integers. Our first result is an explicit formula for the correlation kernel that allows us to analyze the process as N -> infinity. In that limit we obtain a new general class of kernels describing the local correlations close to the equidistant starting points. As the distance between the starting points goes to infinity, the correlation kernel converges to that of a single random walker. As the distance to the starting line increases, however, the local correlations converge to the sine kernel. Thus, this class interpolates between the sine kernel and an ensemble of independent particles. We also compute the scaled simultaneous limit, with both the distance between particles and the distance to the starting line going to infinity, and obtain a process with number variance saturation, previously studied by Johansson.
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| 10. |
- Breuer, Jonathan, et al.
(författare)
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The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles
- 2014
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 265, s. 441-484
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Tidskriftsartikel (refereegranskat)abstract
- We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers under fairly weak conditions on the underlying measure mu. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.
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