A Brief historical introduction to random matrix theory Satya N. - PowerPoint PPT Presentation

A Brief historical introduction to random matrix theory Satya N. Majumdar Laboratoire de Physique Th´ eorique et Mod` eles Statistiques,CNRS, Universit´ e Paris-Sud, France S.N. Majumdar A Brief historical introduction to random matrix theory

First Appearence of Random Matrices S.N. Majumdar A Brief historical introduction to random matrix theory

Covariance Matrix math phys. in general 1 X X 12 11 (MxN) X = X 22 X X 21 2 X 3 X 31 32 in general X 11 X 21 X 31 X t = (NxM) X 12 X X 22 32 2 2 2 W= X t X = X 11 + X 21 + X 31 X 11 X 12 + X 21 X 22 + X 31 X 32 2 + X 2 2 + X 22 X 22 X 21 + X + X 31 X 12 X 12 X 11 32 32 (unnormalized) (NxN) COVARIANCE MATRIX S.N. Majumdar A Brief historical introduction to random matrix theory

Covariance Matrix math phys. in general 1 X X 12 11 (MxN) X = X X 22 X 21 2 X 3 X 31 32 in general X 11 X 21 X 31 X t = (NxM) X 12 X X 22 32 2 2 2 W= X t X = X 11 + X 21 + X 31 X 11 X 12 + X 21 X 22 + X 31 X 32 2 + X 2 2 + X 22 X 22 X 21 + X + X 31 X 12 X 12 X 11 32 32 (unnormalized) (NxN) COVARIANCE MATRIX Null model → random data: X → random ( M × N ) matrix → W = X t X → random N × N matrix (Wishart, 1928) S.N. Majumdar A Brief historical introduction to random matrix theory

RMT in Nuclear Physics: Eugene Wigner S.N. Majumdar A Brief historical introduction to random matrix theory

S.N. Majumdar A Brief historical introduction to random matrix theory

Random Matrices in Nuclear Physics spectra of heavy nuclei �� 238 U E 232 Th E WIGNER (’50) : replace complex H by random matrix DYSON, GAUDIN, MEHTA, ..... S.N. Majumdar A Brief historical introduction to random matrix theory

Applications of Random Matrices Physics: nuclear physics, quantum chaos, disorder and localization, mesoscopic transport, optics/lasers, quantum entanglement, neural networks, gauge theory, QCD, matrix models, cosmology, string theory, statistical physics (growth models, interface, directed polymers...), cold atoms,.... Mathematics: Riemann zeta function (number theory), free probability theory, combinatorics and knot theory, determinantal points processes, integrable systems, ... Statistics: multivariate statistics, principal component analysis (PCA), image processing, data compression, Bayesian model selection, ... Information Theory: signal processing, wireless communications, .. Biology: sequence matching, RNA folding, gene expression network ... Economics and Finance: time series analysis,.... Recent Ref: The Oxford Handbook of Random Matrix Theory ed. by G. Akemann, J. Baik and P. Di Francesco (2011) S.N. Majumdar A Brief historical introduction to random matrix theory

Focus of this course: physical applications Random Matrices and Cold Atoms S.N. Majumdar A Brief historical introduction to random matrix theory

�� Ultracold atoms in a harmonic trap Recent great progress in the experimental manipulation of cold atoms ⇒ to investigate the interplay between quantum and statistical behaviors in many-body systems at low temperatures S.N. Majumdar A Brief historical introduction to random matrix theory

Ultracold atoms in a harmonic trap Recent great progress in the experimental manipulation of cold atoms ⇒ to investigate the interplay between quantum and statistical behaviors in many-body systems at low temperatures A common feature of these experiments ⇒ presence of a confining harmonic potential that traps the particles within a limited spatial region harmonic trap �� x S.N. Majumdar A Brief historical introduction to random matrix theory

Fermi microscope “Quantum-Gas Microscope for Fermionic Atoms”, L.W. Cheuk et. al., PRL, 114, 193001 (2015) S.N. Majumdar A Brief historical introduction to random matrix theory

Spinless Fermi gas in a harmonic potential harmonic trap A particularly simple system: 1-d spinless Fermions at T = 0 in a harmonic trap �� x L −L S.N. Majumdar A Brief historical introduction to random matrix theory

Spinless Fermi gas in a harmonic potential harmonic trap A particularly simple system: 1-d spinless Fermions at T = 0 in a harmonic trap �� x L −L Observable of interest: No. of particles in a box [ − L , L ] � L ˆ n ( x ) ≡ c † ( x ) c ( x ) N L = dx ˆ n ( x ) where ˆ − L S.N. Majumdar A Brief historical introduction to random matrix theory

A Brief historical introduction to random matrix theory Satya N. - PowerPoint PPT Presentation

A Brief historical introduction to random matrix theory Satya N. Majumdar Laboratoire de Physique Th eorique et Mod` eles Statistiques,CNRS, Universit e Paris-Sud, France S.N. Majumdar A Brief historical introduction to random matrix

on Random Matrix Theory MFM Modeling Workshop 2016 Final Report Introduction The purpose of

Numerical analysis and random matrix theory Tom Trogdon ttrogdon@math.uci.edu UC Irvine

Lecture 2. Random Matrix Theory and Phase Transitions of PCA Yuan Yao Hong Kong University of

Expected Value Theory James H. Steiger Department of Psychology and Human Development Vanderbilt

From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014

Random Matrix Theory in a nutshell and applications Manuela Girotti IFT 6085, February 27th,

Large-Nc gauge theory and Chiral Random matrix theory Masanori Hanada Hana Da Masa Nori (KEK

Seeking Gold in Sand Applying Random Matrix Theory to Separation of Signals from Noise in Stock

Cleaning correlation matrices, Random Matrix Theory & HCIZ integrals J.P Bouchaud with: M.

Random matrix models and applications S. Hikami Mathematical and Theoretical Physics Unit OIST

Seeking Gold in Sand Financial applications of Random Matrix Theory in stock market data Mike S.

Random Matrix Theory and applications in cosmology Sbastien Renaux-Petel CNRS - Institut

Matrix-valued Chernoff Bounds and Applications China Theory Week Anastasios Zouzias University

Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian

Random Matrix Theory For Stochastic Structural Dynamics S Adhikari School of Engineering,

Uncertainty Propagation in Linear Systems: An Exact Solution Using random Matrix Theory S

Uncertainty Quantification Using Random Matrix Theory S Adhikari and D Tartakovsky (UCSD) School

Multiplicative chaos in random matrix theory and related fields Christian Webb Aalto University,

A Tale of Two Theories: A Tale of Two Theories: Reconciling Reconciling random matrix theory

Response Variability of Linear Stochastic Systems: A General Solution Using Random Matrix Theory

Chaos, Random Matrix Theory and Spectral Properties of the SYK Model Jacobus Verbaarschot

q fermions or bosons drawn from static random distribution the advent of embedded random

Matrix-correlated random variables: A statistical physics and signal processing duet Florian

YET ANOTHER EASILY FORMULATED YET UNSOLVED PROBLEM IN MATRIX THEORY DMITRY S.

A Brief historical introduction to random matrix theory Satya N. - PowerPoint PPT Presentation

A Brief historical introduction to random matrix theory Satya N. Majumdar Laboratoire de Physique Th eorique et Mod` eles Statistiques,CNRS, Universit e Paris-Sud, France S.N. Majumdar A Brief historical introduction to random matrix

on Random Matrix Theory MFM Modeling Workshop 2016 Final Report Introduction The purpose of

Numerical analysis and random matrix theory Tom Trogdon ttrogdon@math.uci.edu UC Irvine

Lecture 2. Random Matrix Theory and Phase Transitions of PCA Yuan Yao Hong Kong University of

Expected Value Theory James H. Steiger Department of Psychology and Human Development Vanderbilt

From the mesoscopic to microscopic scale in random matrix theory P. Bourgade December 4, 2014

Random Matrix Theory in a nutshell and applications Manuela Girotti IFT 6085, February 27th,

Large-Nc gauge theory and Chiral Random matrix theory Masanori Hanada Hana Da Masa Nori (KEK

Seeking Gold in Sand Applying Random Matrix Theory to Separation of Signals from Noise in Stock

Cleaning correlation matrices, Random Matrix Theory &amp; HCIZ integrals J.P Bouchaud with: M.

Random matrix models and applications S. Hikami Mathematical and Theoretical Physics Unit OIST

Seeking Gold in Sand Financial applications of Random Matrix Theory in stock market data Mike S.

Random Matrix Theory and applications in cosmology Sbastien Renaux-Petel CNRS - Institut

Matrix-valued Chernoff Bounds and Applications China Theory Week Anastasios Zouzias University

Random Matrix Theory Proves that Deep Learning Representations of GAN-data Behave as Gaussian

Random Matrix Theory For Stochastic Structural Dynamics S Adhikari School of Engineering,

Uncertainty Propagation in Linear Systems: An Exact Solution Using random Matrix Theory S

Uncertainty Quantification Using Random Matrix Theory S Adhikari and D Tartakovsky (UCSD) School

Multiplicative chaos in random matrix theory and related fields Christian Webb Aalto University,

A Tale of Two Theories: A Tale of Two Theories: Reconciling Reconciling random matrix theory

Response Variability of Linear Stochastic Systems: A General Solution Using Random Matrix Theory

Chaos, Random Matrix Theory and Spectral Properties of the SYK Model Jacobus Verbaarschot

q fermions or bosons drawn from static random distribution the advent of embedded random

Matrix-correlated random variables: A statistical physics and signal processing duet Florian

YET ANOTHER EASILY FORMULATED YET UNSOLVED PROBLEM IN MATRIX THEORY DMITRY S.

Cleaning correlation matrices, Random Matrix Theory & HCIZ integrals J.P Bouchaud with: M.