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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
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
S.N. Majumdar A Brief historical introduction to random matrix theory
11
X X21 X31 X22 X
phys. math 1 2 3
X =
in general (MxN)
Xt =
X 11 X21 X31 X12
22
X
in general (NxM)
W= XtX =
X11+ X21+ X31
2 2 2
X11X12+ X21 X X22+ X31X X12 X12X11 + X X22X21+ X X31 X12
2 + X22 2 + X2
(unnormalized) COVARIANCE MATRIX (NxN)
32 32 32 32 32 S.N. Majumdar A Brief historical introduction to random matrix theory
11
X X21 X31 X22 X
phys. math 1 2 3
X =
in general (MxN)
Xt =
X 11 X21 X31 X12
22
X
in general (NxM)
W= XtX =
X11+ X21+ X31
2 2 2
X11X12+ X21 X X22+ X31X X12 X12X11 + X X22X21+ X X31 X12
2 + X22 2 + X2
(unnormalized) COVARIANCE MATRIX (NxN)
32 32 32 32 32
Null model → random data: X → random (M × N) matrix → W = X tX → random N × N matrix (Wishart, 1928)
S.N. Majumdar A Brief historical introduction to random matrix theory
S.N. Majumdar A Brief historical introduction to random matrix theory
S.N. Majumdar A Brief historical introduction to random matrix theory
238
232
WIGNER (’50) DYSON, GAUDIN, MEHTA, .....
: replace complex H by random matrix
S.N. Majumdar A Brief historical introduction to random matrix theory
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
S.N. Majumdar A Brief historical introduction to random matrix theory
S.N. Majumdar A Brief historical introduction to random matrix theory
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 Brief historical introduction to random matrix theory
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
S.N. Majumdar A Brief historical introduction to random matrix theory
“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
harmonic trap −L L
A particularly simple system: 1-d spinless Fermions at T = 0 in a harmonic trap
S.N. Majumdar A Brief historical introduction to random matrix theory
harmonic trap −L L
A particularly simple system: 1-d spinless Fermions at T = 0 in a harmonic trap Observable of interest: No. of particles in a box [−L, L] ˆ NL = L
−L
dx ˆ n(x) where ˆ n(x) ≡ c†(x)c(x)
S.N. Majumdar A Brief historical introduction to random matrix theory
harmonic trap −L L
A particularly simple system: 1-d spinless Fermions at T = 0 in a harmonic trap Observable of interest: No. of particles in a box [−L, L] ˆ NL = L
−L
dx ˆ n(x) where ˆ n(x) ≡ c†(x)c(x) Ground state statistics of ˆ NL: Mean: ¯ NL ≡ 0|ˆ NL|0, Variance: VN(L) ≡ 0|ˆ N2
L|0 − ¯
NL
2 Calabrese, Mintchev & Vicari 2011, Vicari 2012, Eisler & Racz 2013, Eisler 2013, Eisler & Peschel 2014
S.N. Majumdar A Brief historical introduction to random matrix theory
S.N. Majumdar A Brief historical introduction to random matrix theory
Question: Can one compute VN(L) vs. L analytically for all L?
S.N. Majumdar A Brief historical introduction to random matrix theory
Variance VN(L) as a function of box size L, for fixed large N
S.N. Majumdar A Brief historical introduction to random matrix theory
Variance VN(L) as a function of box size L, for fixed large N VN(L) ∼
2 βπ2 ln
3 2
for N−1 << L < √ 2 ∼ ˜ Vβ(s), for L = √ 2 +
s √ 2 N− 2
3
∼ exp [−βNφ(L)] for L > √ 2 where φ(L) and ˜ Vβ(s) (for β = 2) are computed explicitly
[R. Marino, S.M., G. Schehr, P. Vivo, PRL 112, 254101 (2014)]
S.N. Majumdar A Brief historical introduction to random matrix theory
S.N. Majumdar A Brief historical introduction to random matrix theory
Number variance VN(L) at T=0:
Entanglement entropy at T = 0:
Generalisation to finite T:
D.S. Dean, P. Le Doussal, S.N.M., & G. Schehr, Phys. Rev. Lett. 110402 (2015)
Generalisation of to higher dimensions
D.S. Dean, P. Le Doussal, S.N.M., & G. Schehr, arXiv:1505.01543
S.N. Majumdar A Brief historical introduction to random matrix theory