scholarly journals SU(N) Wigner–Racah algebra for the matrix of second moments of embedded Gaussian unitary ensemble of random matrices

2005 ◽  
Vol 46 (3) ◽  
pp. 033514 ◽  
Author(s):  
V. K. B. Kota
Keyword(s):  
Random Matrices ◽  
Gaussian Unitary Ensemble ◽  
Second Moments ◽  
The Matrix ◽  
Unitary Ensemble ◽  
Racah Algebra

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

Random growth models

2018 ◽  
Author(s):  
Satya N. Majumdar
Keyword(s):  
Random Matrices ◽  
Growth Model ◽  
Growth Models ◽  
Exclusion Process ◽  
Fixed Time ◽  
Gaussian Unitary Ensemble ◽  
The Family ◽  
Asymmetric Simple Exclusion ◽  
Largest Eigenvalues ◽  
Unitary Ensemble

This article discusses the connection between a particular class of growth processes and random matrices. It first provides an overview of growth model, focusing on the TASEP (totally asymmetric simple exclusion process) with parallel updating, before explaining how random matrices appear. It then describes multi-matrix models and line ensembles, noting that for curved initial data the spatial statistics for large time t is identical to the family of largest eigenvalues in a Gaussian Unitary Ensemble (GUE multi-matrix model. It also considers the link between the line ensemble and Brownian motion, and whether this persists on Gaussian Orthogonal Ensemble (GOE) matrices by comparing the line ensembles at fixed position for the flat polynuclear growth model (PNG) and at fixed time for GOE Brownian motions. Finally, it examines (directed) last passage percolation and random tiling in relation to growth models.

Introduction and guide to the handbook

2018 ◽  
Author(s):  
Gernot Akemann ◽  
Jinho Baik ◽  
Philippe Di Francesco
Keyword(s):  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
The Other ◽  
Random Operators ◽  
New Developments ◽  
Gaussian Unitary Ensemble ◽  
Mathematical Properties ◽  
Unitary Ensemble

This article discusses random matrix theory (RMT) in a nutshell — what it is about, what its main features are, and why it is so successful in applications. It first considers the simplest and maybe most frequently used standard example, the Gaussian Unitary Ensemble (GUE) of random matrices, before looking at several types of applications of RMT, focusing on random operators, counting devices, and RMT without matrices. It then provides a guide to the handbook, explaining how the other forty-two articles on mathematical properties and applications of random matrices are related and built one upon the other. It also lists some topics that are not covered in detail in the book and reviews recent new developments since the first edition of this handbook before concluding with a brief survey of the existing introductory literature.

scholarly journals A PSEUDO-UNITARY ENSEMBLE OF RANDOM MATRICES, PT-SYMMETRY AND THE RIEMANN HYPOTHESIS

2006 ◽  
Vol 21 (04) ◽  
pp. 331-338 ◽  
Author(s):  
ZAFAR AHMED ◽  
SUDHIR R. JAIN
Keyword(s):  
Random Matrices ◽  
Riemann Hypothesis ◽  
Level Spacing ◽  
Gaussian Unitary Ensemble ◽  
Spacing Distribution ◽  
Hamiltonian Connected ◽  
Spectral Interpretation ◽  
Riemann Zeta ◽  
Unitary Ensemble ◽  
Level Spacing Distribution

An ensemble of 2×2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian unitary ensemble found by Wigner. By a re-interpretation of Connes' spectral interpretation of the zeros of Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian).

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