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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/ at arXive)

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1. Introduction

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H = The object S(N×N)

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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

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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

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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?

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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, 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

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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 A n ( ) ckck

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

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Calculating A n ( ) (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+ -D N - -D N + 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 singular contribution from V( ) singularity in density of states (e.g. at edges)

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

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