scholarly journals Random permutations and related topics

2018 ◽  
Author(s):  
Jon Keating ◽  
Nina Snaith
Keyword(s):  
Random Matrix ◽  
Symmetric Spaces ◽  
Matrix Theory ◽  
Irreducible Representations ◽  
Plancherel Measure ◽  
Random Permutations ◽  
Compact Type ◽  
Matrix Groups ◽  
Set Of Permutations ◽  
Random Matrix Ensembles

This article considers some topics in random permutations and random partitions highlighting analogies with random matrix theory (RMT). An ensemble of random permutations is determined by a probability distribution on Sn, the set of permutations of [n] := {1, 2, . . . , n}. In many ways, the symmetric group Sn is linked to classical matrix groups. Ensembles of random permutations should be given the same treatment as random matrix ensembles, such as the ensembles of classical compact groups and symmetric spaces of compact type with normalized invariant measure. The article first describes the Ewens measures, virtual permutations, and the Poisson-Dirichlet distributions before discussing results related to the Plancherel measure on the set of equivalence classes of irreducible representations of Sn and its consecutive generalizations: the z-measures and the Schur measures.

scholarly journals Filling up complex spectral regions through non-Hermitian disordered chains

2022 ◽  
Author(s):  
Hui Jiang ◽  
Ching Hua Lee
Keyword(s):  
Complex Plane ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Quantum Circuits ◽  
High Tolerance ◽  
Stochastic Nature ◽  
Random Matrix Ensembles ◽  
Matrix Ensembles ◽  
Very High

Abstract Eigenspectra that fill regions in the complex plane have been intriguing to many, inspiring research from random matrix theory to esoteric semi-infinite bounded non-Hermitian lattices. In this work, we propose a simple and robust ansatz for constructing models whose eigenspectra fill up generic prescribed regions. Our approach utilizes specially designed non-Hermitian random couplings that allow the co-existence of eigenstates with a continuum of localization lengths, mathematically emulating the effects of semi-infinite boundaries. While some of these couplings are necessarily long-ranged, they are still far more local than what is possible with known random matrix ensembles. Our ansatz can be feasibly implemented in physical platforms such as classical and quantum circuits, and harbors very high tolerance to imperfections due to its stochastic nature.

Random matrices and loop equations

2018 ◽  
Author(s):  
Bertrand Eynard
Keyword(s):  
Random Matrix ◽  
Matrix Theory ◽  
Spectral Curve ◽  
Coulomb Gas ◽  
Algebraic Equations ◽  
Saddle Point Approximation ◽  
Large N ◽  
Topological Recursion ◽  
Random Matrix Ensembles ◽  
Coulomb Gas Method

This chapter is an introduction to algebraic methods in random matrix theory (RMT). In the first section, the random matrix ensembles are introduced and it is shown that going beyond the usual Wigner ensembles can be very useful, in particular by allowing eigenvalues to lie on some paths in the complex plane rather than on the real axis. As a detailed example, the Plancherel model is considered from the point of RMT. The second section is devoted to the saddle-point approximation, also called the Coulomb gas method. This leads to a system of algebraic equations, the solution of which leads to an algebraic curve called the ‘spectral curve’ which determines the large N expansion of all observables in a geometric way. Finally, the third section introduces the ‘loop equations’ (i.e., Schwinger–Dyson equations associated with matrix models), which can be solved recursively (i.e., order by order in a semi-classical expansion) by a universal recursion: the ‘topological recursion’.

Non-Hermitian ensembles

2018 ◽  
Author(s):  
Alexei Morozov
Keyword(s):  
Random Matrix ◽  
Matrix Theory ◽  
Ginibre Ensemble ◽  
Fold Family ◽  
Large Matrix ◽  
Complex Eigenvalues ◽  
Wide Range ◽  
Random Matrix Ensembles ◽  
Matrix Ensembles ◽  
Physical Phenomena

This article discusses the three-fold family of Ginibre random matrix ensembles (complex, real, and quaternion real) and their elliptic deformations. It also considers eigenvalue correlations that are exactly reduced to two-point kernels in the strongly and weakly non-Hermitian limits of large matrix size. Ginibre introduced the complex, real, and quaternion real random matrix ensembles as a mathematical extension of Hermitian random matrix theory. Statistics of complex eigenvalues are now used in modelling a wide range of physical phenomena. After providing an overview of the complex Ginibre ensemble, the article describes random contractions and the complex elliptic ensemble. It then examines real and quaternion-real Ginibre ensembles, along with real and quaternion-real elliptic ensembles. In particular, it analyses the kernel in the elliptic case as well as the limits of strong and weak non-Hermiticity.

scholarly journals Random matrix theory and symmetric spaces

2004 ◽  
Vol 394 (2-3) ◽  
pp. 41-156 ◽  
Author(s):  
M. Caselle ◽  
U. Magnea
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Symmetric Spaces ◽  
Matrix Theory

scholarly journals Fluctuations of the spectrum in rotationally invariant random matrix ensembles

2020 ◽  
pp. 2150025
Author(s):  
Elizabeth S. Meckes ◽  
Mark W. Meckes
Keyword(s):  
Central Limit ◽  
Random Matrix ◽  
Limit Theorems ◽  
Convergence Rates ◽  
Matrix Theory ◽  
Central Limit Theorems ◽  
Inner Product ◽  
Random Matrix Ensembles ◽  
Real Linear ◽  
Rotationally Invariant

We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert–Schmidt inner product) within a real-linear subspace of the space of [Formula: see text] matrices. The matrices, we consider may be real or complex, and Hermitian, antihermitian, or general. We use Stein’s method to prove multivariate central limit theorems, with convergence rates, for these traces of powers, which imply central limit theorems for polynomial linear eigenvalue statistics. In contrast to the usual situation in random matrix theory, in our approach general, nonnormal matrices turn out to be easier to study than Hermitian matrices.

scholarly journals Hurwitz and the origins of random matrix theory in mathematics

2017 ◽  
Vol 06 (01) ◽  
pp. 1730001 ◽  
Author(s):  
Persi Diaconis ◽  
Peter J. Forrester
Keyword(s):  
Invariant Measure ◽  
Explicit Form ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Euler Angles ◽  
Subsequent Work ◽  
Matrix Groups ◽  
The Matrix ◽  
Invariant Group

The purpose of this paper is to put forward the claim that Hurwitz’s paper [Über die Erzeugung der invarianten durch integration, Nachr. Ges. Wiss. Göttingen 1897 (1897) 71–90.] should be regarded as the origin of random matrix theory in mathematics. Here Hurwitz introduced and developed the notion of an invariant measure for the matrix groups [Formula: see text] and [Formula: see text]. He also specified a calculus from which the explicit form of these measures could be computed in terms of an appropriate parametrization — Hurwitz chose to use Euler angles. This enabled him to define and compute invariant group integrals over [Formula: see text] and [Formula: see text]. His main result can be interpreted probabilistically: the Euler angles of a uniformly distributed matrix are independent with beta distributions (and conversely). We use this interpretation to give some new probability results. How Hurwitz’s ideas and methods show themselves in the subsequent work of Weyl, Dyson and others on foundational studies in random matrix theory is detailed.

Random matrix representations of critical statistics

2018 ◽  
Author(s):  
Alexei Borodin
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory ◽  
Luttinger Liquid ◽  
Matrix Representations ◽  
Level Statistics ◽  
Random Matrix Models ◽  
Matrix Ensemble ◽  
Spectral Statistics ◽  
Random Matrix Ensembles

This article examines two random matrix ensembles that are useful for describing critical spectral statistics in systems with multifractal eigenfunction statistics: the Gaussian non-invariant ensemble and the invariant random matrix ensemble. It first provides an overview of non-invariant Gaussian random matrix theory (RMT) with multifractal eigenvectors and invariant random matrix theory (RMT) with log-square confinement before discussing self-unfolding and not self-unfolding in invariant RMT. It then considers a non-trivial unfolding and how it changes the form of the spectral correlations, along with the appearance of a ghost correlation dip in RMT and Hawking radiation. It also describes the correspondence between invariant and non-invariant ensembles and concludes by introducing a simple field theory in 1+1 dimensions which reproduces level statistics of both of the two random matrix models and the classical Wigner-Dyson spectral statistics in the framework of the unified formalism of Luttinger liquid.

SSA, Random Matrix Theory, and Noise-Reduced Correlations

2016 ◽  
Author(s):  
Jan W Dash ◽  
Xipei Yang ◽  
Mario Bondioli ◽  
Harvey J. Stein
Keyword(s):  
Random Matrix Theory ◽  
Random Matrix ◽  
Matrix Theory
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