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  • 1.
    Berggren, Tomas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Mesoscopic Fluctuations for the Thinned Circular Unitary Ensemble2017In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 20, no 3, article id 19Article in journal (Refereed)
    Abstract [en]

    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.

  • 2. Borodin, Alexei
    et al.
    Duits, Maurice
    CALTECH, Dept Math, Pasadena.
    Limits of determinantal processes near a tacnode2011In: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, ISSN 0246-0203, Vol. 47, no 1, p. 243-258Article in journal (Refereed)
    Abstract [en]

    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.

  • 3. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients2017In: Journal of The American Mathematical Society, ISSN 0894-0347, E-ISSN 1088-6834, Vol. 30, no 1, p. 27-66Article in journal (Refereed)
    Abstract [en]

    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.

  • 4. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Nonintersecting paths with a staircase initial condition2012In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 17, p. 1-24Article in journal (Refereed)
    Abstract [en]

    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.

  • 5. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 265, p. 441-484Article in journal (Refereed)
    Abstract [en]

    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.

  • 6. Breuer, Jonathan
    et al.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universality of Mesoscopic Fluctuations for Orthogonal Polynomial Ensembles2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 342, no 2, p. 491-531Article in journal (Refereed)
    Abstract [en]

    We prove that the fluctuations of mesoscopic linear statistics for orthogonal polynomial ensembles are universal in the sense that two measures with asymptotic recurrence coefficients have the same asymptotic mesoscopic fluctuations (under an additional assumption on the local regularity of one of the measures). The convergence rate of the recurrence coefficients determines the range of scales on which the limiting fluctuations are identical. Our main tool is an analysis of the Green's function for the associated Jacobi matrices. As a particular consequence we obtain a central limit theorem for the modified Jacobi Unitary Ensembles on all mesoscopic scales.

  • 7. Delvaux, Steven
    et al.
    Duits, Maurice
    Katholieke Univ Leuven, Belgium.
    An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices2009In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 31, no 4, p. 1894-1914Article in journal (Refereed)
    Abstract [en]

    We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices with rational symbol as the size of the matrix goes to infinity. Our main result is that the weak limit of the normalized eigenvalue counting measure is a particular component of the unique solution to a vector equilibrium problem. Moreover, we show that the other components describe the limiting behavior of certain generalized eigenvalues. In this way, we generalize recent results by Kuijlaars and one of the authors [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 173-196] that were concerned with banded Toeplitz matrices.

  • 8.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gaussian Free Field in an Interlacing Particle System with Two Jump Rates2013In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, no 4, p. 600-643Article in journal (Refereed)
    Abstract [en]

    We study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two-dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field.

  • 9.
    Duits, Maurice
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Painlevé Kernels in Hermitian Matrix Models2014In: Constructive approximation, ISSN 0176-4276, E-ISSN 1432-0940, Vol. 39, no 1, p. 173-196Article in journal (Refereed)
    Abstract [en]

    After reviewing the Hermitian one-matrix model, we will give a brief introduction to the Hermitian two-matrix model and present a summary of some recent results on the asymptotic behavior of the two-matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a 4x4 Riemann-Hilbert problem related to the Painlev, II equation. Also an open problem will be presented.

  • 10.
    Duits, Maurice
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Geudens, D.
    A critical phenomenon in the two-matrix model in the quartic/quadratic case2013In: Duke mathematical journal, ISSN 0012-7094, E-ISSN 1547-7398, Vol. 162, no 8, p. 1383-1462Article in journal (Refereed)
    Abstract [en]

    We study a critical behavior for the eigenvalue statistics in the two-matrix model in the quartic/quadratic case. For certain parameters, the eigenvalue distribution for one of the matrices has a limit that vanishes like a square root in the interior of the support. The main result of the paper is a new kernel that describes the local eigenvalue correlations near that critical point. The kernel is expressed in terms of a 4×4 Riemann-Hilbert problem related to the Hastings-McLeod solution of the Painlevé II equation. We then compare the new kernel with two other critical phenomena that appeared in the literature before. First, we show that the critical kernel that appears in case of quadratic vanishing of the limiting eigenvalue distribution can be retrieved from the new kernel by means of a double scaling limit. Second, we briefly discuss the relation with the tacnode singularity in noncolliding Brownian motions that was recently analyzed. Although the limiting density in that model also vanishes like a square root at a certain interior point, the process at the local scale is different from the process that we obtain in the two-matrix model.

  • 11.
    Duits, Maurice
    et al.
    CALTECH, Pasadena.
    Geudens, Dries
    Kuijlaars, Arno B. J.
    A vector equilibrium problem for the two-matrix model in the quartic/quadratic case2011In: Nonlinearity, ISSN 0951-7715, E-ISSN 1361-6544, Vol. 24, no 3, p. 951-993Article in journal (Refereed)
    Abstract [en]

    We consider the two sequences of biorthogonal polynomials (p(k,n))(k=0)(infinity) and (q(k,n))(k=0)(infinity) related to the Hermitian two-matrix model with potentials V (x) = x(2)/2 and W(y) = y(4)/4 + ty(2). From an asymptotic analysis of the coefficients in the recurrence relation satisfied by these polynomials, we obtain the limiting distribution of the zeros of the polynomials p(n,n) as n -> infinity. The limiting zero distribution is characterized as the first measure of the minimizer in a vector equilibrium problem involving three measures which for the case t = 0 reduces to the vector equilibrium problem that was given recently by two of us. A novel feature is that for t < 0 an external field is active on the third measure which introduces a new type of critical behaviour for a certain negative value of t.

  • 12.
    Duits, Maurice
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Kuijlaars, A.B.J.
    Mo, M. Y.
    The hermitian two matrix model with an even quartic potential2012In: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 217, no 1022, p. 1-118Article in journal (Refereed)
    Abstract [en]

    We consider the two matrix model with an even quartic potential W(y) = y 4/4+ ay 2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1· Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

  • 13.
    Duits, Maurice
    et al.
    Katholieke Univ Leuven, Dept Math.
    Kuijlaars, Arno B. J.
    Katholieke Univ Leuven, Dept Math.
    An equilibrium problem for the limiting eigenvalue distribution of banded Toeplitz matrices2008In: SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, E-ISSN 1095-7162, Vol. 30, no 1, p. 173-196Article in journal (Refereed)
    Abstract [en]

    We study the limiting eigenvalue distribution of n x n banded Toeplitz matrices as n -> infinity. From classical results of Schmidt, Spitzer, and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as n -> infinity. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimizes an energy functional defined on admissible vectors of measures. In addition, we show that each of the other components is the limiting measure of the normalized counting measure on certain generalized eigenvalues.

  • 14.
    Duits, Maurice
    et al.
    Katholieke Univ Leuven, Belgium.
    Kuijlaars, Arno B., J.
    Universality in the two-matrix model: A Riemann-Hilbert Steepest-Descent Analysis2009In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 62, no 8, p. 1076-1153Article in journal (Refereed)
    Abstract [en]

    The eigenvalue statistics of a pair (M(1), M(2)) of n x n Hermitian matrices taken randomly with respect to the measure 1/Z(n) exp (-n Tr(V(M(1)) + W(M(2)) - tau M(1)M(2)))dM(1) dM(2) can be described in terms of two families of biorthogonal polynomials. In this paper we give a steepest-descent analysis of a 4 x 4 matrix-valued Riemann-Hilbert problem characterizing one of the families of biorthogonal polynomials in the special case W(y) = y(4)/4 and V an even polynomial. As a result, we obtain the limiting behavior of the correlation kernel associated to the eigenvalues of M(1) (when averaged over M(2)) in the global and local regime as n -> infinity in the one-cut regular case. A special feature in the analysis is the introduction of a vector equilibrium problem involving both an external field and an upper constraint.

1 - 14 of 14
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