Random Matrix: From Wigner to Quantum
Chaos
Horng-Tzer Yau
Harvard University
Joint work with
P. Bourgade, L. Erd˝
os, B. Schlein and J. Yin
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Perhaps I am now too courageous when I try to
guess the distribution of the distances between successive levels (of energies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the
problem in a simpleminded fashion.The question is
simply what are the distances of the characteristic
values of a symmetric matrix with random coefficients.
Eugene Wigner, 1956
Nobel prize 1963
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Classical Example of Universality—-Central Limit Theorem
Gaussian distribution with variance σ 2:
x2
exp −
µ(x) = √
2
2σ 2
2πσ
Suppose Xj , j = 1, . . . , N are independent random variables with
mean zero and variance one. Then the probability distribution
of
X1 + . . . + XN
√
N
converges to the Gaussian distribution with variance one. Independence assumption can be weaken, but is crucial.
1
4
Model for independent particles, Poisson statistics: Law of
putting K particles independently in an interval of size N so that
K/N → ρ is fixed.
Question: How to model systems with high correlation?
Gaussian unitary ensemble: H = (hjk )1≤j,k≤N hermitian with
2
1
√
√
(xjk + iyjk )
and
hjj =
xjj
hjk =
N
N
and where xjk , yjk , j > k, and xjj are independent centered Gaussian random variables with variance 1/2.
Classical ensembles: Gaussian unitary ensemble (GUE), Gaussian orthogonal ensemble (GOE), Gaussian symplectic ensemble(GSE), sample covariance ensembles (Important for statistics
applications).
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• E. Wigner (1955): The excitation spectra of heavy nuclei
have the same spacing distribution as the eigenvalues of GOE.
Experimental data for excitation spectra of heavy nuclei:
typical Poisson statistics:
Typical random matrix eigenvalues
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Global Statistics: Density of state ρ(x) follows the Wigner semicircle law for GUE, GOE and GSE.
ρ (x) =
−2
1
2π
4 − x2
2
Eigenvalues: λ1 ≤ λ2 ≤ . . . . . . λN , λi+1 − λi ∼ 1/N .
Moment method: for all k fixed:
1
1 2 k
k
TrH →
x 4 − x2dx
N
2π −2
Z
q
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Wigner surmise (1956):
"
the gap distribution ∼
πx2
πx
exp −
2
4
#
dx
Idea: guess the probability law from the eigenvalues of 2 × 2
matrices.
Correct up to a few percentage points when compared with the
GOE gap distribution.
8
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Probability density of eigenvalues (w.r.t. Lebesgue measure)
pN (λ1, ..., λN )
Correlation function for two eigenvalues:
Z
(2)
pN (x1, x2) =
pN (x1, x2, ..., xN )dx3...dxN
RN −2
Density of states:
ρN (x) =
Z
RN −1
pN (x, x2, ..., xN )dx2...dxN
Dyson, Gaudin, Mehta [’60]: local statistics of level correlation
a
a
(2)
1
2
,E +
,
lim [ρN (E)]−2pN E +
N →∞
N ρN (E)
N ρN (E)
sin πa
(for GUE), |E| < 2
πa
Spacing distribution can be computed from the correlation functions.
o2
= det S(ai − aj )
,
i,j=1
n
S(a) =
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Quantum Chaos conjecture
Energy levels in quantum billiards or H = −∆ + V .
Laplace equation:
−∆ψn = λnψn.
Related Question: Random Schrodinger equation
V is random by P. Anderson 1958
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Berry-Tabor conjecture (1977):If the billiard trajectories are
integrable, the eigenvalue spacings statistics is given by the
Poisson distribution e−xdx.
Bohigas-Giannoni-Schmit conjecture (1984):If the billiard is chaotic,
the eigenvalue spacings statistics is given by the GOE.
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Other application of random matrices:
tions, biology, finance, traffic
Wireless communica-
Wishart (1928), Statistics application: Sample covariance ensembles, matrix of the form A+A, A is the data matrix.
• Riemann ζ-function: Gap distribution of zeros of ζ function
is given by GUE (Montgomery, 1973). Odlyzko data:
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There are essentially two different behaviors in nature:
A: Poisson statistics, for systems with little or no correlations.
B: Random matrix statistics: for systems with high correlations.
(Edge behavior is different from the bulk.)
Fundamental belief of universality of random matrices: The
macroscopic statistics depend on the models, but the microscopic statistics are independent of the details of the systems
except the symmetries.
Classical many-body systems are too hard to solve; Boltzmann’s
model of classical statistical physics e−βH .
Quantum many-body systems (and highly correlated systems)
are too hard to solve; Wigner’s model of random matrices.
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Unitary ensemble: Hermitian matrices with density
P(H)dH ∼ e−βN Tr V (H)dH
Invariant under H → U HU −1 for any unitary U (GUE)
p(λ1, . . . , λN ) ∼
Y
−β
(λi − λj )β e
P
j
N V (λj )
∼ e−N βHN
i<j
X1
1 X
V (λi) −
log |λj − λi|
HN =
N i<j
i 2
classical ensembles β = 1, 2, 4 GOE, GUE, GSE.
The distribution of the density of λj is given by the equilibrium
measure ρV (x)dx which is the minimizer of
I(ν) =
Z
V (t)dν(t) −
ZZ
log |t − s|dν(s)dν(t)
The semicircle law is the case V (x) = x2.
Suppose the support of ρV = [A, B].
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From VanderMonde determinant structure (β = 2),
(2)
pN (x1, x2) = C det[KN (xi, xj )]2
i,j=1
√
KN (x, y) =


ψ (x)ψN −1(y) − ψN (y)ψN −1(x) 
.
N N
x−y
ψN orthogonal polynomials w.r.t. e−βV (x)dx.
large N asymptotic of orthogonal polynomials =⇒ local eigenvalue statistics indep of V . But density of e.v. depends on V .
KN (x + a1, x + a2) → S(a1 − a2),
S(a) =
sin πa
(for GUE),
πa
Dyson (1962-76), Gaudin-Mehta (1960- ) via Hermite polynomials and general cases by Deift-Its-Zhou (1997), Bleher-Its
(1999), Deift-et al (1999-2009) [Riemann-Hilbert method], PasturSchcherbina (1997), Lubinsky (2008) . . .
β = 1, 4:(Widom), Deift-Gioev, Kriecherbauer-Shcherbina: Assuming V analytic with some additional assumptions (convex).
From 1960 to 2008, all results depend on explicit formulas.
Why need different methods for different cases?
non-classical β?
what about
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Generalized Wigner Ensembles
H = (hkj )1≤k,j≤N ,
2,
Ehij = 0, E|hij |2 = σij
¯
hji = hij
X
independent
2 = 1, σ ∼ N −1/2 .
σij
ij
i
c
2 ≤ C
≤ σij
N
N
If hij are i.i.d. then it is called Wigner ensembles.
(A)
Universality conjecture (Wigner-Dyson-Mehta conjecture):
If hij are independent, then the local eigenvalues statistics are
the same as the Gaussian ensembles. No results up to 2008.
More generally, eigenvalue gap distributions depends only on
symmetry classes and are independent of models.
It is the same for both invariant and non-invariant models (but
depend on β which is a symmetry parameter).
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Theorem [Erd˝
os, Knowles, Schlein, Y, Yin ] The bulk universality holds for generalized Wigner ensembles satisfying (A) and
E|xij |4+ε ≤ C,
0
then for −2 < E < 2, b = N −1+ε , ε0 > 0
Z E+b
dE 0 (k)
b
b
(k)
pF,N − pµ,N E 0 + 1 , . . . , E 0 + k = 0
lim
N →∞ E−b 2b
N
N
weakly
F
µ
generalized symmetric matrices
GOE
generalized hermitian
GUE
generalized self-dual quaternion
GSE
real covariance
real Gaussian Wishart
complex covariance
complex Gaussian Wishart
Variances can vary in this theorem. Comparison with Tao-Vu
later.
Generalization to Erd˝
os-Reyni graphs by Knowles-Erd˝
os-Y-Yin
(some example of quantum chaos with random data)
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Theorem[Bourgade-Erdoes-Y 2010-2011] Suppose that V is real
analytic and
V 00(x) ≥ −C
(1)
Consider the β-ensemble µβ,V with β > 0. Let E ∈ (A, B) and
E 0 ∈ (−2, 2). We have, as N → ∞,
α1
1
α2
(2)
p
x+
,x +
%V (E)2 V,N
N %(E)
N %(E)
α
α
1
(2)
1
2
* 0.
p
x
+
,
x
+
−
G,N
0
2
0
0
%G(E )
N %G(E )
N %G(E )
G stands for the Gaussian with V (x) = x2.
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Three Steps to the Universality: Non-invariant case
Step 1. A priori estimate, Local Semicircle Law.
Method: System of self-consistent equations for the Green function, control the error by large deviation methods.
Step 2. Universality of Gaussian divisible ensembles
√
√
t > 0, H0 is Wigner V is GUE
H = 1 − tH0 + tV,
i.e., the matrix entries have some Gaussian components.
General method based on estimating the convergence to local
equilibrium of Dyson Brownian motion.
Step 3. Approximation by Gaussian divisible ensembles—–
A density argument. Resolvent perturbation expansion to remove
the Gaussian part in step 2.
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Two Steps to the Universality: Invariant case
Step 1: A priori estimate (similar to the Step 1 in the noninvariant case)
Theorem [Rigidity estimate] For any α > 0 and > 0, there is a
constant c > 0 such that for any N ≥ 1 and k ∈ JαN, (1 − α)N K,
Pµβ,V
|λk − γk | > N −1+
c
≤ ce−cN .
(Valid only in the bulk.)
Key input: Logarithmic Sobolev inequality, Loop equation.
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Step 2. Uniqueness of log gas.
We order the particles and study the statistics of :
{λj : j ∈ I},
I = IL := JL + 1, L + K K,
Relabeling:
(λ1, λ2, . . . , λN ) := (y1, . . . yL, xL+1, . . . xL+K , yL+K+1, . . . yN )
local equilibrium measure on x with boundary condition y: µV
y (dx).
µV
y (dx) ∼ exp(−N βHy ),
X1
1 X
Hy (x) =
Vy (xi)−
log |xj −xi|
2
N
i,j∈I
i∈I
i<j
2 X
Vy (x) = V (x) −
log |x − yj |.
N j6∈I
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Goal: µy in the bulk is independent of y as N, K → ∞ for ”good
boundary conditions”, i.e., Prove the uniqueness of Gibbs measure for good boundary conditions.
Tool: Study the relaxation to equilibrium for dynamics with µG
y as
the invariant measure. This is a generalization of Dyson Brownian Motion.
Uniqueness of Gibbs state is established via a dynamical argument, i.e., estimate of relaxation to equilibrium of a ”local
DBM”.
Fundamental reason of Universality for both invariant and noninvariant: fast relaxation to local equilibrium of DBM.
A priori estimate is needed to provide an estimate of local relaxation time.
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Date
07-08
09May
09May
Erdos, Schlein, Y, Yin
Local semicircle N −1 scale
Prior results scale N −1/2
delocalization of eigenvector
Univ. for small N −3/4
Gaussian component v.s.
Johansson’s O(1) Gaussian
Univ. for Hermitian matrix
smooth dist., first univ.
09Jun
09Jun
09Jul
Joint paper, hermitian bulk
Removes all extra conditions
Univ. for symm Wigner case
DBM argument appeared
Tao and Vu
Local semicircle reproved
4 moment thm for e. values
Hermitian bulk univ for
vanishing 3rd moment.
Bernoulli dist. excluded.
univ.
apart from subexp tail of hij
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Date
09Nov
Erdos, Schlein, Y, Yin
Bulk univ for real and
complex covariance matrices
09Dec
10Jan
10Mar
10Jul
11Mar
11Apr
11Dec
Tao and Vu
Bulk univ for complex
cov. matrices reproved.
Bulk univ for Wigner matrices
with varying variances,
except Bernoulli
Green fn comparison thm
Real Bernoulli solved
Eigenvalue rigidity proved
Dyson conjecture for
optimal relaxation proved
univ. for sparse matrices
finite 4 + moments for Wigner
Bulk univ. for general β
ensemble with convex V
Convexity condition removed
Main Open Problems
Universality random Schr¨
odinger
Universality of band matrices
Random band matrices: H is symmetric with independent but
not identically distributed entries with mean zero and variance
E W |hk`|2 = e−|k−`|/W
√
Narrow band, W N =⇒
localization, Poisson statistics
√
Broad band, W N =⇒
delocalization, GOE statistics
Even the Gaussian case is open.
d-regular graphs
Prove some examples of quantum chaos.
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