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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,

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Presentation on theme: "Shohat’s Method and Universality in Random Matrix Theory Weizmann Institute of Science Eugene Kanzieper Department of Condensed Matter Physics Rehovot,"— Presentation transcript:

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2 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

3 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)

4 1. Introduction

5 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)

6 symmetry fixed H  S(N×N) P(H) invariant under appropriate rotation P(S H S -1 )= P(H) ‘cause of trace invariant matrix model doesn’t relate to any dynamic properties of modelled random system but underlying symmetry incorporated properly symmetry becomes manifest in eigenvalue representation P(H)  exp { – Tr V(H) } Joint probability distribution function confinement potential

7 orthogonal ensemble (real symmetric matrix) H † = H T = H unitary ensemble (complex Hermitean matrix) H † = H symplectic ensemble (real quaternion matrix) H † = H = – (1 N   y ) H T (1 N   y )  Cartan’s SS (Altland & Zirnbauer, 1997) : 10 Symmetry classes (Dyson, 1962) 1 2 4

8 P(H)  exp { – Tr V(H) } What is confinement potential? no first principle may fix V(H) statistical independence of H ij : 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?

9 soft edge origin bulk Airy Law Bessel Law Sine Law N ( ) Local correlations at  = 2

10 Pastur (1992) Brezin and Zee (1993) … Nishigaki (1996) Akemann, Damgaard, Magnea, and Nishigaki (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

11 2. Technical Preliminaries and The Strategy

12 Preliminaries - 1  joint probability distribution function  n-point correlation function

13 Preliminaries - 2  two-point kernel  Christoffel-Darboux theorem  three-term recurrence equation  orthonormality

14 The strategy ? !

15 3. Shohat’s Method (1939)

16 Step No 1

17 Step No 2

18 Step No 2 (continued) !

19 Remarks  exact!  useful? … but nonlinear: - not really for more complicated potentials at finite n - ok up to  large-N …

20 Large-N analysis 

21 Calculating A n ( ) ckck

22 Calculating A n ( ) – auxiliary identity (math induction) ckck

23 Calculating A n ( ) (continued) !

24

25 Large-N differential equation

26 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

27 Three kernels for nothing and universality 1) Spectrum bulk and the Sine kernel

28 Three kernels for nothing and universality 2) Spectrum origin and the Bessel kernel

29 Three kernels for nothing and universality 3) Spectrum edge and the Airy kernel

30 Percy Deift’s talk: Two-band random matrices DN-DN- DN+DN+ -D N - -D N + 0

31 4. Conclusions

32 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 … ?

33 a way to get novel correlations: two sources but: care (!) precisely at singularity! direct singular contribution from V( ) singularity in density of states (e.g. at edges)

34 1939 universality might have been well understood in the very early days of RMT …


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