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Shohat’s Method and Universality in Random Matrix Theory

Department of Condensed Matter Physics. Weizmann Institute of Science. Rehovot, Israel. Shohat’s Method and Universality in Random Matrix Theory. Eugene Kanzieper. James H Simons Workshop on Random Matrix Theory, Stony Brook, February 22, 2002. Review. E Kanzieper and V Freilikher

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Shohat’s Method and Universality in Random Matrix Theory

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  1. Department of Condensed Matter Physics Weizmann Institute of Science Rehovot, Israel Shohat’s Method and Universality in Random Matrix Theory Eugene Kanzieper James H Simons Workshop on Random Matrix Theory, Stony Brook, February 22, 2002

  2. 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)

  3. 1. Introduction

  4. The object 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= S(N×N)

  5. Joint probability distribution function P(H)  exp{– TrV(H)} • confinement potential symmetry fixed H S(N×N) • P(H) invariant under appropriate rotation P(SH 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

  6. Symmetry classes (Dyson, 1962)  • orthogonal ensemble (real symmetric matrix) H†= HT = H 1 • unitary ensemble (complex Hermitean matrix) H†= H 2 • symplectic ensemble (real quaternion matrix) H† = H = – (1N y)HT(1N y) 4 • Cartan’s SS (Altland & Zirnbauer, 1997):10

  7. What is confinement potential? P(H)  exp{– TrV(H)} ! • no first principle may fix V(H) • ? statistical independence of Hij: V(H) = H2 (Gaussian ensembles) Universality Problem What is the influence of confinement potential V(H) on (local) eigenvalue correlations? Fox and Kahn (1964); Leff (1964); Bronk (1965)

  8. Local correlations at = 2 Bessel Law Sine Law N () Airy Law bulk origin  soft edge

  9. References (fairly incomplete … ) Sine Pastur (1992) Brezin and Zee (1993) … Bessel Nishigaki (1996) Akemann, Damgaard, Magnea, and Nishigaki (1997) … = 2 Airy Bowick and Brezin (1991) Kanzieper and Freilikher (1997) … = 1 and 4 Other symmetry classes: Tracy and Widom (1998), Widom (1999) Sener and Verbaarschot (1998) Klein and Verbaarchot (2000) …

  10. 2. Technical Preliminaries and The Strategy

  11. Preliminaries - 1  joint probability distribution function  n-point correlation function

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

  13. The strategy ? !

  14. 3. Shohat’s Method (1939)

  15. Step No 1

  16. Step No 2

  17. Step No 2 (continued) !

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

  19. Large-N analysis

  20. Calculating An() ck

  21. Calculating An() –auxiliary identity ck (math induction)

  22. Calculating An() (continued) !

  23. Calculating An() (continued)

  24. Large-N differential equation

  25. 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

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

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

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

  29. Percy Deift’s talk: Two-band random matrices 0 DN- -DN+ -DN- DN+

  30. 4. Conclusions

  31. • launched the Shohat’s method in RMT context • essence: mapping 3-term recurrence onto 2nd order differential equation (large-N behaviourof 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 … ?

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

  33. 1939 universality might have been well understood in the very early days of RMT …

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