shohat’s method and universality in random matrix theory weizmann institute of science eugene...
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Shohat’s Method and Universality in Random Matrix Theory
Weizmann Institute of Science
Eugene Kanzieper
Department of Condensed Matter Physics
Rehovot, Israel
James H Simons Workshop on Random Matrix Theory, Stony Brook, February 22, 2002
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Review
E Kanzieper and V Freilikher Spectra of large random matrices: A method of study
In
Diffuse Waves in Complex Media (ed. J-P Fouque)
NATO ASI, Series C (Mathematical and Physical Sciences)
Vol 531, pp 165 – 211 (Kluwer, 1999)
(cond-mat/9809365 at arXive)
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1. Introduction
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0.28 2.40 1.39 5.73 4.28 0.18 9.33 4.58 9.27 7.30 4.03 4.05 1.59 6.49 9.19
2.63 5.03 6.25 4.78 8.45 0.02 9.52 6.97 4.20 1.14 9.93 5.94 6.49 5.03 4.50
2.94 4.78 4.98 6.41 4.02 0.01 5.17 9.32 4.73 3.00 3.19 0.74 8.03 4.38 1.30
7.24 8.04 0.39 1.83 2.47 8.03 6.60 4.34 9.47 9.93 5.94 6.49 4.78 4.85 3.28
9.45 4.82 4.06 4.06 7.37 9.03 8.05 4.51 3.95 4.00 3.05 3.58 7.10 4.48 9.37
4.86 5.07 7.35 4.78 8.45 0.02 9.52 6.97 4.20 8.03 7.94 5.29 1.18 4.38 3.01
1.27 8.13 5.37 0.09 5.32 3.86 8.22 0.36 0.88 0.28 2.40 1.39 6.60 4.34 9.47
8.03 7.94 5.29 1.18 2.87 1.14 9.93 5.94 6.49 4.78 8.45 0.02 9.52 6.97 4.20
6.73 4.18 4.96 3.00 5.29 3.57 5.29 8.83 7.17 2.40 1.39 5.73 4.28 0.18 9.33
9.52 6.97 4.20 0.28 2.40 1.39 5.73 6.41 4.02 0.01 5.17 5.07 7.35 4.78 8.45
5.07 7.35 4.78 8.45 7.30 4.03 4.05 1.59 6.49 9.19 3.02 4.39 4.04 9.03 8.10
6.60 4.34 9.47 9.93 5.94 6.49 4.78 4.85 3.28 7.24 8.04 0.39 1.83 2.47 8.03
9.33 4.58 9.27 7.30 4.03 4.05 1.59 6.49 9.19 2.63 5.03 6.25 4.78 8.45 0.02
7.17 2.40 1.39 5.73 4.28 0.18 9.33 9.52 6.97 4.20 0.28 2.40 1.39 5.73 6.41
4.02 0.01 5.17 5.07 7.35 4.78 8.45 5.07 7.35 4.78 8.45 7.30 4.03 4.05 1.59
H =
The object
S(N×N)
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symmetry fixed H S(N×N)•P(H) invariant under appropriate rotationP(S H S-1)= P(H) ‘cause of trace
•
invariant matrix model doesn’t relate to any dynamic properties of modelled random systembut underlying symmetry incorporated properly
•
symmetry becomes manifest in eigenvalue representation
•
P(H) exp{– Tr V(H)}
Joint probability distribution function
confinement potential
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orthogonal ensemble (real symmetric matrix)H† = HT = H
•
unitary ensemble (complex Hermitean matrix)H† = H
•
symplectic ensemble (real quaternion matrix)H† = H = – (1N
y) HT (1N y)
•
Cartan’s SS (Altland & Zirnbauer, 1997): 10•
Symmetry classes (Dyson, 1962)
1
2
4
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P(H) exp{– Tr V(H)}
What is confinement potential?
no first principle may fix V(H)•statistical independence of Hij: V(H) = H
2
(Gaussian ensembles)•
!?
Fox and Kahn (1964); Leff (1964); Bronk (1965)
Universality Problem
What is the influence of confinement potential V(H) on (local) eigenvalue
correlations?
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soft edge
origin
bulk
Airy Law
Bessel Law Sine Law
N ()
Local correlations at = 2
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Pastur (1992)Brezin and Zee (1993)…
Nishigaki (1996)Akemann, Damgaard, Magnea, andNishigaki (1997)…
Bowick and Brezin (1991) Kanzieper and Freilikher (1997)…
= 2
Other symmetry classes:
Tracy and Widom (1998), Widom (1999)Sener and Verbaarschot (1998)Klein and Verbaarchot (2000)…
= 1 and 4
References (fairly incomplete … )
Sine
Bessel
Airy
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2. Technical Preliminaries
and The Strategy
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Preliminaries - 1
joint probability distribution function
n-point correlation function
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Preliminaries - 2
two-point kernel
Christoffel-Darboux theorem
three-term recurrence equation
orthonormality
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The strategy
?
!
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3. Shohat’s Method (1939)
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Step No 1
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Step No 2
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Step No 2 (continued)
!
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Remarks
exact!
useful?
… but nonlinear:
- not really for more complicated potentials at finite n
- ok up to
large-N …
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Large-N analysis
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Calculating An()
ck
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Calculating An() – auxiliary identity
(math induction)
ck
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Calculating An() (continued)
!
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Calculating An() (continued)
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Large-N differential equation
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Comments
large-N limit
‘mean-field’ approximation for coefficients and
Dyson’s density of states (not always the case!)
- singular contribution out of log- indirect dependence on V otherwise! - stable with respect to deformations of confinement potential
easy generalisations (two allowed bands…)
universality of three kernels for free
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Three kernels for nothing and universality
1) Spectrum bulk and the Sine kernel
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Three kernels for nothing and universality
2) Spectrum origin and the Bessel kernel
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Three kernels for nothing and universality
3) Spectrum edge and the Airy kernel
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Percy Deift’s talk: Two-band random matrices
DN- DN
+-DN--DN
+
0
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4. Conclusions
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launched the Shohat’s method in RMT context
essence: mapping 3-term recurrence onto 2nd order differential equation (large-N behaviour of r-coefficients as input)
demonstrated universality in easy and coherent way
other applications: - global correlators- 2-band random matrices- multicritical correlations at edges
q-deformed ensembles, non-Hermitean RMT … ?
••
•
•
•
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a way to get novel correlations: two sources
but: care (!) precisely at singularity!
•
direct singularcontribution from
V()
singularity in density of states (e.g. at edges)
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1939universality might have been well understood
in the very early days of RMT …