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General beta Jacobi corners process and the Gaussian Free Field
Alexei Borodin,Vadim Gorin +1 more
TLDR
In this article, it was shown that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles.Abstract:
We prove that the two-dimensional Gaussian Free Field describes the asymptotics of global fluctuations of a multilevel extension of the general beta Jacobi random matrix ensembles. Our approach is based on the connection of the Jacobi ensembles to a degeneration of the Macdonald processes that parallels the degeneration of the Macdonald polynomials to to the Heckman-Opdam hypergeometric functions (of type A). We also discuss the beta goes to infinity limit.read more
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Integrable probability: From representation theory to Macdonald processes
Alexei Borodin,Leonid Petrov +1 more
TL;DR: The lecture notes for a mini-course given at the 2013 Cornell Probability Summer School as discussed by the authors discuss the ( q, t )-deformation of those leading to the Macdonald processes, nearest neighbor dynamics, their limit to semi-discrete Brownian polymers and large time asymptotic analysis of polymer's partition function.
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Stochastic six-vertex model
TL;DR: In this paper, the authors study the asymmetric six-vertex model in the quadrant with parameters on the stochastic line and show that the random height function converges to an explicit deterministic limit shape as the mesh size tends to 0.
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Fractional brownian motion with hurst index h=0 and the gaussian unitary ensemble
TL;DR: In this paper, the authors established a relation between characteristic polynomials of N×N GUE random matrices H as N→∞, and Gaussian processes with logarithmic correlations.
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Nearest neighbor Markov dynamics on Macdonald processes
Alexei Borodin,Leonid Petrov +1 more
TL;DR: In this paper, a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes is presented, which unifies known examples of such dynamics and also yields many new ones.
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Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients
Jonathan Breuer,Maurice Duits +1 more
TL;DR: In this article, fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles are studied for which the underlying biorthyogonal family of families is known.
References
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Book
Symmetric functions and Hall polynomials
TL;DR: In this paper, the characters of GLn over a finite field and the Hecke ring of GLs over finite fields have been investigated and shown to be symmetric functions with two parameters.
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Orthogonal Polynomials
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Journal ArticleDOI
On the Distribution of the Roots of Certain Symmetric Matrices
TL;DR: The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
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Log-Gases and Random Matrices
TL;DR: Forrester as discussed by the authors presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems, and provides hundreds of guided exercises and linked topics.
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An Introduction to Random Matrices
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.