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Search Results: 1 - 10 of 3369 matches for " Kurt Johansson "
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Universality of the local spacing distribution in certain ensembles of hermitian Wigner matrices
Kurt Johansson
Physics , 2000, DOI: 10.1007/s002200000328
Abstract: Consider an $N\times N$ hermitian random matrix with independent entries, not necessarily Gaussian, a so called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as $N\to\infty$, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices.
Transversal fluctuations for increasing subsequences on the plane
Kurt Johansson
Physics , 1999,
Abstract: Consider a realization of a Poisson process in R^2 with intensity 1 and take a maximal up/right path from the origin to (N,N) consisting of line segments between the points, where maximal means that it contains as many points as possible. The number of points in such a path has fluctuations of order N^chi, where chi=1/3 by a result of Baik-Deift-Johansson. Here we show that typical deviations of a maximal path from the diagonal x=y is of order N^xi with xi=2/3. This is consistent with the scaling identity chi=2xi-1, which is believed to hold in many random growth models.
Non-intersecting Paths, Random Tilings and Random Matrices
Kurt Johansson
Physics , 2000,
Abstract: We investigate certain measures induced by families of non-intersecting paths in domino tilings of the Aztec diamond, rhombus tilings of an abc-hexagon, a dimer model on a cylindrical brick lattice and a growth model. The measures obtained, e.g. the Krawtchouk and Hahn ensembles, have the same structure as the eigenvalue measures in random matrix theory like GUE, which can in fact be obtained from non-intersecting Brownian motions. The derivations of the measures are based on the Karlin-McGregor or Lindstr\"om-Gessel-Viennot method. We use the measure to show some asymptotic results for the models.
Toeplitz determinants, random growth and determinantal processes
Kurt Johansson
Mathematics , 2003,
Abstract: We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory.
From Gumbel to Tracy-Widom
Kurt Johansson
Mathematics , 2005,
Abstract: The Tracy-Widom distribution that has been much studied in recent years can be thought of as an extreme value distribution. We discuss interpolation between the classical extreme value distribution $\exp(-\exp(-x))$, the Gumbel distribution and the Tracy-Widom distribution. There is a family of determinantal processes whose edge behaviour interpolates between a Poisson process with density $\exp(-x)$ and the Airy kernel point process. This process can be obtained as a scaling limit of a grand canonical version of a random matrix model introduced by Moshe, Neuberger and Shapiro. We also consider the deformed GUE ensemble, $M=M_0+\sqrt{2S} V$, with $M_0$ diagobal with independent elements and $V$ from GUE. Here we do not see a transition from Tracy-Widom to Gumbel, but rather a transition from Tracy-Widom to Gaussian.
Random matrices and determinantal processes
Kurt Johansson
Mathematics , 2005,
Abstract: We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.
A multi-dimensional Markov chain and the Meixner ensemble
Kurt Johansson
Mathematics , 2007, DOI: 10.1007/s11512-008-0089-6
Abstract: We show that the transition probability of the Markoc chain $(G(j,1),...,G(j,n))_{j\ge 1}$, 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.
Shape Fluctuations and Random Matrices
Kurt Johansson
Mathematics , 1999, DOI: 10.1007/s002200050027
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Discrete polynuclear growth and determinantal processes
Kurt Johansson
Mathematics , 2002, DOI: 10.1007/s00220-003-0945-y
Abstract: We consider a discrete polynuclear growth (PNG) process and prove a functioal limit theorem for its convergrence to the Airy process. This generalizes previous results by Pr"ahofer and Spohn. The result enables us to express the GOE largest eigenvalue (Tracy-Widom) distribution in terms of the Airy process. We also show some results and give a conjecture about the transversal fluctuations in a point to line last passage percolation problem.
The arctic circle boundary and the Airy process
Kurt Johansson
Mathematics , 2003, DOI: 10.1214/009117904000000937
Abstract: We prove that the, appropriately rescaled, boundary of the north polar region in the Aztec diamond converges to the Airy process. The proof uses certain determinantal point processes given by the extended Krawtchouk kernel. We also prove a version of Propp's conjecture concerning the structure of the tiling at the center of the Aztec diamond.
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