Random numbers and random matrices: Quantum chaos meets number theory

2006 ◽  
Vol 74 (6) ◽  
pp. 547-553 ◽  
Author(s):  
Todd Timberlake
Keyword(s):  
Number Theory ◽  
Random Matrices ◽  
Quantum Chaos ◽  
Random Numbers

scholarly journals Random matrices and quantum chaos

2001 ◽  
Vol 98 (19) ◽  
pp. 10531-10532 ◽  
Author(s):  
T. Kriecherbauer ◽  
J. Marklof ◽  
A. Soshnikov
Keyword(s):  
Random Matrices ◽  
Quantum Chaos

scholarly journals UNIVERSAL CORRELATIONS IN RANDOM MATRICES: QUANTUM CHAOS, THE 1/r2 INTEGRABLE MODEL, AND QUANTUM GRAVITY

1996 ◽  
Vol 11 (15) ◽  
pp. 1201-1219 ◽  
Author(s):  
SANJAY JAIN
Keyword(s):  
Quantum Gravity ◽  
Random Matrices ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Correlation Functions ◽  
Quantum Chaos ◽  
Matrix Theory ◽  
Mathematical Formulation ◽  
Integrable Model ◽  
Moser System

Random matrix theory (RMT) provides a common mathematical formulation of distinct physical questions in three different areas: quantum chaos, the 1-D integrable model with the 1/r2 interaction (the Calogero-Sutherland-Moser system) and 2-D quantum gravity. We review the connection of RMT with these areas. We also discuss the method of loop equations for determining correlation functions in RMT, and smoothed global eigenvalue correlators in the two-matrix model for Gaussian orthogonal, unitary and symplectic ensembles.

scholarly journals Random Matrices, Magic Squares and Matching Polynomials

10.37236/1859 ◽  
2004 ◽  
Vol 11 (2) ◽  
Author(s):  
Persi Diaconis ◽  
Alex Gamburd
Keyword(s):  
Random Matrices ◽  
Riemann Zeta Function ◽  
Quantum Chaos ◽  
Higher Moments ◽  
Characteristic Polynomials ◽  
Stochastic Matrices ◽  
Unitary Matrices ◽  
Magic Squares ◽  
Riemann Zeta ◽  
Random Unitary Matrices

Characteristic polynomials of random unitary matrices have been intensively studied in recent years: by number theorists in connection with Riemann zeta-function, and by theoretical physicists in connection with Quantum Chaos. In particular, Haake and collaborators have computed the variance of the coefficients of these polynomials and raised the question of computing the higher moments. The answer turns out to be intimately related to counting integer stochastic matrices (magic squares). Similar results are obtained for the moments of secular coefficients of random matrices from orthogonal and symplectic groups. Combinatorial meaning of the moments of the secular coefficients of GUE matrices is also investigated and the connection with matching polynomials is discussed.

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