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  • 1. Adler, M.
    et al.
    Chhita, S.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, P.
    Tacnode GUE-minor processes and double Aztec diamonds2015In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 162, no 1-2, p. 275-325Article in journal (Refereed)
    Abstract [en]

    We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.

  • 2. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, Pierre
    Double Aztec diamonds and the tacnode process2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, p. 518-571Article in journal (Refereed)
    Abstract [en]

    Discrete and continuous non-intersecting random processes have given rise to critical "infinite-dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the tacnode process. It is also-shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.

  • 3. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, Pierre
    Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class2018In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 21, no 1, article id 9Article in journal (Refereed)
    Abstract [en]

    This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.

  • 4.
    Adler, Mark
    et al.
    Brandeis Univ, Dept Math, Waltham, MA 02453 USA..
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    van Moerbeke, Pierre
    Univ Louvain, Dept Math, B-1348 Louvain, Belgium.;Brandeis Univ, Waltham, MA 02453 USA..
    Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes2018In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 364, no 1, p. 287-342Article in journal (Refereed)
    Abstract [en]

    This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model, we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.

  • 5. Baik, J.
    et al.
    Deift, P.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    On the distribution of the length of the second row of a young diagram under Plancherel measure2000In: Geometric and Functional Analysis, ISSN 1016-443X, E-ISSN 1420-8970, Vol. 10, no 4, p. 702-731Article in journal (Refereed)
  • 6.
    Beffara, Vincent
    et al.
    Univ Grenoble Alpes, CNRS, Inst Fourier, F-38000 Grenoble, France..
    Chhita, Sunil
    Univ Durham, Dept Math Sci, Durham DH1 3LE, England..
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Royal Inst Technol KTH, Dept Math, Linstedtsvagen 25, SE-10044 Stockholm, Sweden..
    AIRY POINT PROCESS AT THE LIQUID-GAS BOUNDARY2018In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 46, no 5, p. 2973-3013Article in journal (Refereed)
    Abstract [en]

    Domino tilings of the two-periodic Aztec diamond feature all of the three possible types of phases of random tiling models. These phases are determined by the decay of correlations between dominoes and are generally known as solid, liquid and gas. The liquid-solid boundary is easy to define microscopically and is known in many models to be described by the Airy process in the limit of a large random tiling. The liquid-gas boundary has no obvious microscopic description. Using the height function, we define a random measure in the two-periodic Aztec diamond designed to detect the long range correlations visible at the liquid-gas boundary. We prove that this random measure converges to the extended Airy point process. This indicates that, in a sense, the liquid-gas boundary should also be described by the Airy process.

  • 7. Chhita, S.
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Young, B.
    Asymptotic domino statistics in the Aztec diamond2015In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 25, no 3, p. 1232-1278Article in journal (Refereed)
    Abstract [en]

    We study random domino tilings of the Aztec diamond with different weights for horizontal and vertical dominoes. A domino tiling of an Aztec diamond can also be described by a particle system which is a determinantal process. We give a relation between the correlation kernel for this process and the inverse Kasteleyn matrix of the Aztec diamond. This gives a formula for the inverse Kasteleyn matrix which generalizes a result of Helfgott. As an application, we investigate the asymptotics of the process formed by the southern dominoes close to the frozen boundary. We find that at the northern boundary, the southern domino process converges to a thinned Airy point process. At the southern boundary, the process of holes of the southern domino process converges to a multiple point process that we call the thickened Airy point process. We also study the convergence of the domino process in the unfrozen region to the limiting Gibbs measure.

  • 8. Chhita, Sunil
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Domino statistics of the two-periodic Aztec diamond2016In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 294, p. 37-149Article in journal (Refereed)
    Abstract [en]

    Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplification of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond, where the asymptotic analysis is simpler, we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the inverse Kasteleyn matrix at both the ‘liquid–solid’ and ‘liquid–gas’ boundaries, and find the extended Airy kernel in the next order asymptotics. Finally we provide a potential candidate for a combinatorial description of the liquid–gas boundary.

  • 9. Duits, Maurice
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Powers of large random unitary matrices and toeplitz determinants2010In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 362, no 3, p. 1169-1187Article in journal (Refereed)
    Abstract [en]

    We study the limiting behavior of Tr U-k(n), where U is an n x n random unitary matrix and k(n) is a natural number that may vary with n in an arbitrary way. Our analysis is based on the connection with Toeplitz determinants. The central observation of this paper is a strong Szego limit theorem for Toeplitz determinants associated to symbols depending on 71 in a particular way. As a consequence of this result, we find that for each fixed m is an element of N, the random variables Tr U-kj(n)/root min(k(j)(n),n), j = 1,..., m, converge to independent standard complex normals.

  • 10.
    Duse, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Metcalfe, A.
    The Cusp-Airy process2016In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 21, article id 57Article in journal (Refereed)
    Abstract [en]

    At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point but their formula is not completely correct.

  • 11.
    Duse, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Metcalfe, Anthony
    The Cusp-Airy ProcessManuscript (preprint) (Other academic)
    Abstract [en]

    At a typical cusp point of the disordered region in a random tiling model we expect to see a determinantal process called the Pearcey process in the appropriate scaling limit. However, in certain situations another limiting point process appears that we call the Cusp-Airy process, which is a kind of two sided extension of the Airy kernel point process. We will study this problem in a class of random lozenge tiling models coming from interlacing particle systems. The situation was briefly studied previously by Okounkov and Reshetikhin under the name cuspidal turning point.

  • 12.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A multi-dimensional Markov chain and the Meixner ensemble2010In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 48, no 1, p. 79-95Article in journal (Refereed)
    Abstract [en]

    We show that the transition probability of the Markov chain (G(i,1),...,G(i,n)) (ia parts per thousand yen1), where the G(i,j)'s are certain directed last-passage times, is given by a determinant of a special form. An analogous formula has recently been obtained by Warren in a Brownian motion model. Furthermore we demonstrate that this formula leads to the Meixner ensemble when we compute the distribution function for G(m,n). We also obtain the Fredholm determinant representation of this distribution, where the kernel has a double contour integral representation.

  • 13.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Course 1 Random matrices and determinantal processes2006In: Les Houches Summer School Proceedings, 2006, no C, p. 1-56Conference paper (Refereed)
  • 14.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Determinantal processes with number variance saturation2004In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 252, no 3-Jan, p. 111-148Article in journal (Refereed)
  • 15.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Discrete orthogonal polynomial ensembles and the Plancherel measure2001In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 153, no 1, p. 259-296Article in journal (Refereed)
  • 16.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Discrete polynuclear growth and determinantal processes2003In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 242, no 2-Jan, p. 277-329Article in journal (Refereed)
  • 17.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    From Gumbel to Tracy-Widom2007In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 138, no 2-Jan, p. 75-112Article in journal (Refereed)
  • 18.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-colliding Brownian Motions and the Extended Tacnode Process2013In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 319, no 1, p. 231-267Article in journal (Refereed)
    Abstract [en]

    We consider non-colliding Brownian motions with two starting points and two endpoints. The points are chosen so that the two groups of Brownian motions just touch each other, a situation that is referred to as a tacnode. The extended kernel for the determinantal point process at the tacnode point is computed using new methods and given in a different form from that obtained for a single time in previous work by Delvaux, Kuijlaars and Zhang. The form of the extended kernel is also different from that obtained for the extended tacnode kernel in another model by Adler, Ferrari and van Moerbeke. We also obtain the correlation kernel for a finite number of non-colliding Brownian motions starting at two points and ending at arbitrary points.

  • 19.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Non-intersecting paths, random tilings and random matrices2002In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 123, no 2, p. 225-280Article in journal (Refereed)
  • 20.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Non-intersecting, simple, symmetric random walks and the extended Hahn kernel2005In: Annales de l'Institut Fourier, ISSN 0373-0956, E-ISSN 1777-5310, Vol. 55, no 6, p. 2129-2145Article in journal (Refereed)
  • 21.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On some special directed last-passage percolation models2008In: Integrable Systems And Random Matrices: In Honor Of Percy Deift / [ed] Baik, J; Kriecherbauer, T; Li, LC; McLaughlin, KDT; Tomei, C, 2008, Vol. 458, p. 333-346Conference paper (Refereed)
    Abstract [en]

    We investigate extended processes given by last-passage times in directed models defined using exponential variables with decaying mean. In certain cases we find the universal Airy process, but other cases lead to non-universal or trivial extended processes.

  • 22.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI).
    Random matrices and determinantal processes2006In: Mathematical statistical physics / [ed] Edited by Anton Bovier, François Dunlop, Aernout van Enter, Frank den Hollander and Jean Dalibard, Amsterdam: Elsevier, 2006, p. 1-55Chapter in book (Refereed)
  • 23.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Shape fluctuations and random matrices2000In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 209, no 2, p. 437-476Article in journal (Refereed)
  • 24.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The arctic circle boundary and the Airy process2005In: Annals of Probability, ISSN 0091-1798, E-ISSN 2168-894X, Vol. 33, no 1, p. 1-30Article in journal (Refereed)
  • 25.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The two-time distribution in geometric last-passage percolation2019In: Probability Theory and Related FieldsArticle in journal (Refereed)
    Abstract [en]

    We study the two-time distribution in directed last passage percolation with geometric weights in the first quadrant. We compute the scaling limit and show that it is given by a contour integral of a Fredholm determinant. 

  • 26.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Transversal fluctuations for increasing subsequences on the plane2000In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 116, no 4, p. 445-456Article in journal (Refereed)
  • 27.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Two Time Distribution in Brownian Directed Percolation2016In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, p. 1-52Article in journal (Refereed)
    Abstract [en]

    In the zero temperature Brownian semi-discrete directed polymer we study the joint distribution of two last-passage times at positions ordered in the time-like direction. This is the situation when we have the slow de-correlation phenomenon. We compute the limiting joint distribution function in a scaling limit. This limiting distribution is given by an expansion in determinants that is not a Fredholm expansion. A somewhat similar looking formula was derived non-rigorously in a related model by Dotsenko.

  • 28.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Universality for certain Hermitian Wigner matrices under weak moment conditions2012In: Annales de l'I.H.P. Probabilites et statistiques, ISSN 0246-0203, E-ISSN 1778-7017, Vol. 48, no 1, p. 47-79Article in journal (Refereed)
    Abstract [en]

    We study the universality of the local eigenvalue statistics of Gaussian divisible Hermitian Wigner matrices. These random matrices are obtained by adding an independent GUE matrix to an Hermitian random matrix with independent elements, a Wigner matrix. We prove that Tracy-Widom universality holds at the edge in this class of random matrices under the optimal moment condition that there is a uniform bound on the fourth moment of the matrix elements. Furthermore, we show that universality holds in the bulk for Gaussian divisible Wigner matrices if we just assume finite second moments.

  • 29.
    Johansson, Kurt
    KTH, Superseded Departments, Mathematics.
    Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices2001In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 215, no 3, p. 683-705Article in journal (Refereed)
  • 30.
    Lambert, Gaultier
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processesManuscript (preprint) (Other academic)
  • 31.
    Nordenstam, Eric
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Eigenvalues of GUE minors2006In: Electronic Journal of Probability, ISSN 1083-6489, E-ISSN 1083-6489, Vol. 11, no 50, p. 1342-1371Article in journal (Refereed)
    Abstract [en]

    Consider an infinite random matrix H = (hij)o<i,j picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by Hi = (hrs)1≤r,s≤i and let the j:th largest eigenvalue of Hi be μji. We show that the configuration of all these eigenvalues (i, μji) form a determinantal point process on ℕ × ℝ. Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.

1 - 31 of 31
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