scholarly journals EXACTLY SOLVABLE SCALING THEORY OF CONDUCTION IN DISORDERED WIRES

1994 ◽  
Vol 08 (08n09) ◽  
pp. 469-478 ◽  
Author(s):  
C. W. J. Beenakker
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Scaling Theory ◽  
Transmission Eigenvalues ◽  
Conductance Fluctuations ◽  
Recent Developments ◽  
Log Normal ◽  
Universal Conductance ◽  
Conductance Distribution

Recent developments in the scaling theory of phase-coherent conduction through a disordered wire are reviewed. The Dorokhov–Mello–Pereyra–Kumar equation for the distribution of transmission eigenvalues has been solved exactly, in the absence of time-reversal symmetry. Comparison with the previous prediction of random-matrix theory shows that this prediction was highly accurate but not exact: the repulsion of the smallest eigenvalues was overestimated by a factor of two. This factor of two resolves several disturbing discrepancies between random-matrix theory and microscopic calculations, notably in the magnitude of the universal conductance fluctuations in the metallic regime, and in the width of the log-normal conductance distribution in the insulating regime.

scholarly journals Random matrix theory for the analysis of the performance of an analog computer: a scaling theory

2004 ◽  
Vol 323 (3-4) ◽  
pp. 204-209 ◽  
Author(s):  
Asa Ben-Hur ◽  
Joshua Feinberg ◽  
Shmuel Fishman ◽  
Hava T. Siegelmann
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Analog Computer ◽  
Scaling Theory

scholarly journals Random matrix theory and quantum chromodynamics

2018 ◽  
Author(s):  
Gernot Akemann
Keyword(s):  
Quantum Chromodynamics ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Chemical Potential ◽  
Matrix Theory ◽  
Space Time ◽  
Gaussian Unitary Ensemble ◽  
Density Correlation ◽  
Recent Developments ◽  
Matrix Problems

This chapter was originally presented to a mixed audience of physicists and mathematicians with some basic working knowledge of random matrix theory. The first part is devoted to the solution of the chiral Gaussian unitary ensemble in the presence of characteristic polynomials, using orthogonal polynomial techniques. This includes all eigenvalue density correlation functions, smallest eigenvalue distributions, and their microscopic limit at the origin. These quantities are relevant for the description of the Dirac operator spectrum in quantum chromodynamics with three colors in four Euclidean space-time dimensions. In the second part these two theories are related based on symmetries, and the random matrix approximation is explained. In the last part recent developments are covered, including the effect of finite chemical potential and finite space-time lattice spacing, and their corresponding orthogonal polynomials. This chapter also provides some open random matrix problems.

Random matrices and number theory: some recent themes

2018 ◽  
Author(s):  
Jon P. Keating
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gaussian Random Fields ◽  
Extreme Value Statistics ◽  
Matrix Methods ◽  
Recent Developments ◽  
Riemann Zeta

The aim of this chapter is to motivate and describe some recent developments concerning the applications of random matrix theory to problems in number theory. The first section provides a brief and rather selective introduction to the theory of the Riemann zeta function, in particular to those parts needed to understand the connections with random matrix theory. The second section focuses on the value distribution of the zeta function on its critical line, specifically on recent progress in understanding the extreme value statistics gained through a conjectural link to log–correlated Gaussian random fields and the statistical mechanics of glasses. The third section outlines some number-theoretic problems that can be resolved in function fields using random matrix methods. In this latter case, random matrix theory provides the only route we currently have for calculating certain important arithmetic statistics rigorously and unconditionally.

Sign in / Sign up

Export Citation Format

Share Document