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bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Replica method: a statistical mechanics approachto probability-based information processing
Toshiyuki Tanakatt@i.kyoto-u.ac.jp
Graduate School of Informatics, Kyoto University, Kyoto, Japan
17th International Symposium on Mathematical Theory ofNetworks and Systems, Kyoto, Japan, July 25, 2006
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Introduction
Replica method
Developed in studies of spin glasses (=magnetic materialswith random spin-spin interactions)
Recently applied to problems in information sciences:
Neural networksStatistical learning theoryCombinatorial optimization problemsError-correcting codesCDMA (digital wireless communication)Eigenvalue distribution of random matrices
Still lacks rigorous mathematical justification
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Introduction
Objectives
To give a review of the replica method, as well as itsmathematically questionable point.
To demonstrate its applications.
Eigenvalue distribution of random matrices.Analysis of digital communication systems.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Problem
Basic defs.
A: N × N real symmetric random matrixλ1, . . . , λN : Eigenvalues of A
Empirical eigenvalue distribution
ρA(x) =1
N
N∑i=1
δ(x − λi )
Problem
To evaluateρ(x) = lim
N→∞ EA
[ρA(x)
]
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Basic results
Wigner’s semicircle law (Wigner, 1951)
A = (aij): N × N matrix, aij (i ≤ j): i.i.d., mean 0, variance 1/N.
Marc̆enko-Pastur law (Marc̆enko & Pastur, 1967)
A = ΞTΞ, Ξ = (ξμi ): p × N matrix;ξμi i.i.d., mean 0, variance 1/N.
(Girko’s) full-circle law (Girko, 1985)
A = (aij), aij : i.i.d., mean 0, variance 1/N. (A not symmetric)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Wigner’s semicircle law
Wigner’s semicircle law
A = (aij), aij (i ≤ j): i.i.d, mean 0, variance 1/N.
⇒ ρ(x) =
⎧⎨⎩
1
2π
√4 − x2 (|x | < 2)
0 (|x | > 2)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Wigner’s semicircle law
-2 -1 0 1 2
Histogram of eigenvalues of a 6000 × 6000 random symmetricmatrix with entries following Gaussian distribution.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Marc̆enko-Pastur law
Marc̆enko-Pastur law
A = ΞTΞ, Ξ = (ξμi ): p × N matrix;ξμi : i.i.d., mean 0, variance 1/N.
ρ(x) =
⎧⎪⎪⎨⎪⎪⎩
√4α − (x − 1 − α)2
2πxχα(x) (α ≥ 1)
(1 − α)δ(x) +
√4α − (x − 1 − α)2
2πxχα(x) (0 < α < 1)
α ≡ p/Nχα(x): Characteristic function of interval [(1 −√
α)2, (1 +√
α)2].
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Marc̆enko-Pastur law
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 1 2 3 4 5 6
ρ(x)
x
α = 0.3, 0.6, 2; Terms proportional to δ(x) not shown.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Full-circle
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im(λ
)
Re(λ)
Eigenvalue distribution of a real 6000 × 6000 random matrix.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Eigenvalue distribution of random matrices
Useful for what?
Wide applications in mathematical physics
Applications in Information Processing
Statistical learning theoryDigital communication(kernel) PCA (bioinformatics, mathematical finance, etc.)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Eigenvalue distribution of random matrix
Approaches
Marginalization of joint eigenvalue distribution (ex. Mehta,
1967)
Evaluation of moments (ex. Brody et al., 1981)
“Locator” expansion (ex. Bray & Moore, 1979; Hertz et al., 1989)
Cavity method
Free probability theory (ex. Voiculescu, 1985; Hiai & Petz, 2000)
Replica method (ex. Edwards & Jones, 1976)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Reformulation
ρA(x) =1
N
N∑i=1
δ(x − λi)
mA(z) =1
Ntr(A − zI )−1
=2
N
d
dzlog ZA(z)
ZA(z) = (−2πi)N/2|A − zI |−1/2
=
∫R
N
exp[− i
2uT (A − zI )u
]du
mA(z) =
∫R
ρA(x)
x − zdx
ρA(x) = limε→+0
1
π�[
mA(x + iε)]
Stieltjes trans.
tr∗−1 = (log det ∗)′
Gaussian integ.rep. of det.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Averaging over A
ρ(x) = EA
[1
N
N∑i=1
δ(x − λi)
]
m(z) = EA
[1
Ntr(A − zI )−1
]
= 2d
dzEA
[1
Nlog ZA(z)
]
ZA(z) =
∫R
N
exp[− i
2uT (A − zI )u
]du
mA(z) =
∫R
ρA(x)
x − zdx
ρA(x) = limε→+0
1
π�[
mA(x + iε)]
Stieltjes trans.
tr∗−1 = (log det ∗)′
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Outline of the approach
1 Evaluate
f (z) = limN→∞ EA
[1
Nlog ZA(z)
]
where (ZA(z) =
∫R
N
exp[− i
2uT (A − zI )u
]du
)
2 Calculate Stieltjes transform m(z) of ρ(x) with
m(z) = 2d
dzf (z).
3 Evaluate the inverse Stieltjes transform to obtain ρ(x):
ρ(x) = limε→+0
1
π�[
m(x + iε)]
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Replica method
Rewriting of formulas
f (z) = limN→∞ EA
[1
Nlog ZA(z)
](
∂
∂nlog EA
(Z n
)=
EA(Z n log Z )
EA(Z n)
)
= limN→∞
1
Nlimn→0
∂
∂nlog EA
{[ZA(z)
]n}(
Exchange order of limn→0 ∂/∂nand limN→∞.
)
= limn→0
∂
∂nlim
N→∞1
Nlog EA
{[ZA(z)
]n}
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Replica method
Goal
To evaluate limN→∞
1
Nlog EA
{[ZA(z)
]n}.
The replica “trick”
Evaluate it by assuming n to be a positive integer.
Believe the result to be valid for real n.
No mathematically rigorous justification.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Random matrix ensemble
Covariance matrix of random samples
A = ΞTΞ, Ξ = (ξμi ){ξμi ; μ = 1, . . . , p; i = 1, . . . , N} i.i.d.,
E(ξ) = 0, E(ξ2) = O(1/N), E(ξm) = o(1/N) (m ≥ 3)⇒ Marc̆enko-Pastur
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Replica method
Introduction of “Replicated” systems
ZA(z) =
∫R
N
exp[− i
2uT (A − zI )u
]du
⇒ For n = 1, 2, . . .,
[ZA(z)
]n=
∫R
Nn
exp[− i
2
n∑a=1
uTa (A − zI )ua
] n∏a=1
dua
EA
{[ZA(z)
]n}=
∫R
NnEA
[exp
(− i
2
n∑a=1
uTa Aua
)]
× exp( iz
2
n∑a=1
∣∣ua
∣∣2) n∏a=1
dua
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Calculation of EA(· · · )Factor to be evaluated
EA
[exp
(− i
2
n∑a=1
uTa Aua
)]
Assumptions on ramdom mtx. ensemble
A = ΞTΞ, Ξ = (ξμi ) (size: p × N)
{ξμi ; μ = 1, . . . , p; i = 1, . . . , N}: i.i.d.
vμa ≡N∑
i=1
ξμiuai ⇒ uTa Aua =
p∑μ=1
(vμa
)2
n∑a=1
uTa Aua =
p∑μ=1
n∑a=1
(vμa
)2=
p∑μ=1
∣∣vμ
∣∣2, (vμ = (vμ1, . . . , vμn)
T)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Calculation of EA(· · · )Average over A = Average over v
vμa ≡N∑
i=1
ξμiuai ⇒ uTa Aua =
p∑μ=1
(vμa
)2
n∑a=1
uTa Aua =
p∑μ=1
n∑a=1
(vμa
)2=
p∑μ=1
∣∣vμ
∣∣2(vμ = (vμ1, . . . , vμn)
T)
⇒ EA
[exp
(− i
2
n∑a=1
uTa Aua
)]=
{Ev
[exp
(− i
2
∣∣v∣∣2)]}p
(v = (ξTu1, . . . , ξTun)
T)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Assumptions on ramdom mtx. ensemble
E(ξ) = 0, E(ξ2) = 1/N, E(ξm) = o(1/N) (m ≥ 3)
Statistical properties of v = (ξTu1, . . . , ξTun)T
v ∼ N(0, Q) for fixed {ua} (⇐ Central limit theorem)
Order parameters
Q = (qab), qab ≡ Eξ(vvT ) = N−1
N∑i=1
uaiubi
⇒ Ev
[exp
(− i
2
∣∣v∣∣2)]=
∣∣I + iQ∣∣−1/2
(Gaussian integral)
exp( iz
2
n∑a=1
∣∣ua
∣∣2) = exp(Niz
2trQ
)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Integral with {ua} = Integral with Q
Ev
[exp
(− i
2
∣∣v∣∣2)]=
∣∣I + iQ∣∣−1/2
(Gaussian integral)
exp( iz
2
n∑a=1
∣∣ua
∣∣2) = exp(Niz
2trQ
)
⇒ EA
{[ZA(z)
]n}=
∫eNG(Q) μ(Q) dQ
G(Q) ≡ −α
2log
∣∣I + iQ∣∣ +
iz
2trQ, α ≡ p/N
μ(Q) ≡∫ ∏
a≤b
δ
(qab − 1
N
N∑i=1
uaiubi
) n∏a=1
dua
(Subshell volume)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
Varadhan’s theorem (Large deviation theory)
limN→∞
1
Nlog
∫eNG(Q) μ(Q) dQ = sup
Q
[G(Q) − I(Q)]
Rate function I(Q)
The heuristic formula μ(Q) = e−NI(Q) holds for large N with
I(Q) = −1
2log |Q| + n
2
[1 + log(−2π)
].
Stationary condition (saddle-point equation)
∂
∂Q
[G(Q) − I(Q)]
= O ⇒ izI + Q−1 − iα(I + iQ)−1 = O
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic resultsApproachesAnalysis via replica method
m(z) = limn→0
∂
∂nitrQ, izI + Q−1 − iα(I + iQ)−1 = O
Q uniquely determined by requiringintegrals not to diverge.
⎧⎪⎪⎨⎪⎪⎩
Q = qI
iz +1
q− iα
1 + iq= 0
m(z) = iq
⇒ m(z) = −[z − α
1 + m(z)
]−1
ρ(x) =
⎧⎪⎪⎨⎪⎪⎩
√4α − (x − 1 − α)2
2πxχα(x) (α ≥ 1)
(1 − α)δ(x) +
√4α − (x − 1 − α)2
2πxχα(x) (0 < α < 1)
(Marc̆enko-Pastur)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
CommunicationBasic setting
X
Input
Channel Y
Output
p(x) p(y |x)
Output Y · · · What we observe.
Input X · · · What we want to know!
Problem
How much the output Y conveys information about the input X?
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Communication
Problem
How much the output Y conveys information about the input X?
Mutual information
I (X ; Y ) =
∫p(x , y) log
p(x , y)
p(x)p(y)dx dy
= H(Y ) − H(Y |X )
H(Y |X ) = −∫ [∫
p(y |x) log p(y |x) dy
]p(x) dx
H(Y ) = −∫
p(y) log p(y) dy
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Use of randomness in communication
Basic diagram
X
Input
Channel? Y
Output
Error-correcting code: ? =Encoder
Modulation: ? =Modulator
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Use of randomness in communication
Examples
“Turbo” codes (Berrou et al., 1993): Two convolutionalcodes interlinked with a random interleaver.
Low-density parity-check codes (Gallager, 1962): Randomensemble of low-density parity-check matrices.
Code-division multiple-access (CDMA): Spreadingmodulation with random spreading sequences.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Mobile communication
�
�
��
�
�
�
�
�
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Multiple access and CDMA
Multiple-access: Multipleusers simultaneously commu-nicate with the same basestation.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
2-user CDMA system
×
⇑
×
⇑
Noise
×
⇓
⇓
Alice
Information x1
Spreading code{sμ1}
Bob
Information x2
Spreading code{sμ2}
Base st.
{yμ} Rcvd. signal
{sμ1}
h1 ⇒ x̂1
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
K -user CDMA system
x1
x2
xK
�
�
�
Information
{sμ1}{sμ2} {sμK}� � �
Spreading codes
××
×
�
�
�
Channel
++
Noise{nμ}
Rcvd. signal{yμ}
yμ =1√N
K∑k=1
sμkxk + nμ
(μ = 1, . . . , N)
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Use of randomness in communication
Basic diagram
X
Input
ChannelS Y
Output
p(x) p(y |x , s)
Modeling of randomness
p(y |x , s): Channel input-output characteristics depending onauxiliary random variable S (e.g., spreading codes).
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Use of randomness in communication
Randomness-averaged mutual information
One wants to evaluate:
ES
[I (X ; Y |S)
]= ES
[H(Y |S)
] − ES
[H(Y |X , S)
]
Difficulty
ES
[H(Y |S)
]= −ES
[∫p(y |S) log p(y |S) dy
]
p(y |S) =
∫p(y |x , S) p(x) dx
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Replica method
Evaluation of randomness-averaged entropy
ES
[H(Y |S)
]= −ES
[∫p(y |S) log p(y |S) dy
]= − lim
n→0
∂
∂nlog Ξn
Ξn ≡ ES
[∫ [p(y |S)
]n+1dy
]=
∫∫ [p(y |s)]n+1
p(s) dy ds
Replica method provides a powerful approach to evaluaterandomness-averaged mutual information.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Example of results
10-710-610-510-410-310-210-1100
0 2 4 6 8 10 12
Bit-
Err
or R
ate
Pb
Signal-to-Noise Ratio Eb /N0 [dB]
System load β = 1, 1.2, 1.4, 1.6, 1.8, 2Single-user limit
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Discussion
S-shaped performance curve
Multiple solutions for performance.
Essentially the same as magnetization curve of ferromagnets:{Stable, Metastable, Unstable} solutions
-1
-0.5
0
0.5
1
-0.4 -0.2 0 0.2 0.4
Mag
netiz
atio
n
External magnetic field
“Hysteresis” · · · Affects behavior of estimation algorithms.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Replica method: Applications
Digital communication
Turbo codes (Montanari & Sourlas, 2000)
Low-density parity-check codes (Murayama et al., 2000)
Code-division multiple-access (Tanaka, 2002)
· · ·
Other fields
Associative memory of neural network, Hopfield model (Amitet al., 1985)
Perceptron learning (Gardner & Derrida, 1989)
Random K -SAT problem (Monasson & Zecchina, 1997)
· · ·
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Basic settingUse of randomness in communicationReplica method
Replica method
Mathematics
Validity of replica solutions (not replica method) (Talagrand, 2003)
“It is difficult to see (in the replica method) more than away to guess the correct formula.” — Talagrand, 2003.
Current status
Empirically gives the correct results to various problems.
Heuristics: Validity unknown.
Justification (or counterexample) needed.
Toshiyuki Tanaka MTNS2006: Replica method
bg=black!2 IntroductionLimiting eigenvalue distribution of random matrices
Digital communicationBibliography
Bibliography
D. J. Amit et al., Phys. Rev. Lett., 55(14), 1530–1533, 1985.
Berrou et al., Proc. IEEE Int. Conf. Commun., 1064–1070, 1993.
A. Bray & Moore, J. Phys. C: Solid State Phys., 12(11), L441–L448, 1979.
T. A. Brody et al., Rev. Mod. Phys., 53(3), 385–479, 1981.
S. F. Edwards & R. C. Jones, J. Phys. A: Math. Gen., 9(10), 1595–1603, 1976.
R. G. Gallager, Trans. IRE Info. Theory, 8, 21–28, 1962.
E. Gardner & B. Derrida, J. Phys. A: Math. Gen., 22(12), 1983–1994, 1989.
V. L. Girko, Theory of Prob. Its Appl. (USSR), 29(4), 694–706, 1985.
J. A. Hertz et al., J. Phys. A: Math. Gen., 22(12), 2133–2150, 1989.
F. Hiai & D. Petz, The Semicircle Law, Free Random Variables, and Entropy, Amer. Math. Soc., 2000.
V. A. Marc̆enko & L. A. Pastur, Math. USSR-Sb., 1, 457–483, 1967.
M. L. Mehta, Random Matrices, Academic Press, 1967.
R. Monasson & R. Zecchina, Phys. Rev. E, 56(2), 1357–1370, 1997.
A. Montanari & N. Sourlas, Eur. Phys. J. B, 18(1) 107–119, 2000.
T. Murayama et al., Phys. Rev. E, 62(2), 1577–1591, 2000.
M. Talagrand, Spin Glasses: A Challenge for Mathematicians, Springer, 2003.
T. Tanaka, IEEE Trans. Info. Theory, 48(11), 2888–2910, 2002.
D. V. Voiculescu, in Operator Algebras and Their Connection with Topology and Ergodic Theory,Lecture Notes in Math., 1131, Springer, 1985, 556–588.
E. P. Wigner, Proc Cambridge Phil. Soc., 47, 790–798, 1951.
Toshiyuki Tanaka MTNS2006: Replica method
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