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

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)

1. Introduction

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)

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

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

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?

soft edge

origin

bulk

Airy Law

Bessel Law Sine Law

N ()

Local correlations at = 2

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

2. Technical Preliminaries

and The Strategy

Preliminaries - 1

joint probability distribution function

n-point correlation function

Preliminaries - 2

two-point kernel

Christoffel-Darboux theorem

three-term recurrence equation

orthonormality

The strategy

?

!

3. Shohat’s Method (1939)

Step No 1

Step No 2

Step No 2 (continued)

!

Remarks

exact!

useful?

… but nonlinear:

- not really for more complicated potentials at finite n

- ok up to

large-N …

Large-N analysis

Calculating An()

ck

Calculating An() – auxiliary identity

(math induction)

ck

Calculating An() (continued)

!

Calculating An() (continued)

Large-N differential equation

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

Three kernels for nothing and universality

1) Spectrum bulk and the Sine kernel

Three kernels for nothing and universality

2) Spectrum origin and the Bessel kernel

Three kernels for nothing and universality

3) Spectrum edge and the Airy kernel

Percy Deift’s talk: Two-band random matrices

DN- DN

+-DN--DN

+

0

4. Conclusions

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 … ?

••

•

•

•

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)

1939universality might have been well understood

in the very early days of RMT …