Application of the $tau$-Function Theory of Painlev'e Equations to Random Matrices PIV, PII

≤j<k≤N
|λk − λj |2 ,
(1.1)
C denoting the normalization. (Throughout the symbol C will be used to denote some constant, i.e. a quantity 2 independent of the primary variable(s) of the equation.) The choice g (x) = e−x , which is realized by choosing √ each diagonal element of X independently from the normal distribution N[0, 1/ 2], and each off diagonal independently with distribution N[0, 1/2] + iN[0, 1/2], is referred to as the Gaussian Unitary Ensemble (GUE) and is the main focus of the present article. Specifically our interest is in the distribution of the largest eigenvalue, and the average values of powers (integer and fractional) of the characteristic polynomial N l=1 (λ − λl ) for such matrices. Let EN (0; (s, ∞)) denote the probability that there are no eigenvalues in the interval (s, ∞) for N × N GUE matrices. The distribution of the largest eigenvalue pmax (s) is given in terms of EN (0; (s, ∞)) by pmax (s) = d EN (0; (s, ∞)). ds (1.2)
Contents 1. Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. τ -Function Theory for PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 3.5 Affine Weyl group symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toda lattice equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B¨ acklund transformations and discrete Painlev´ e systems . . . . . . . . . . . . . . . . . . . . . . . Affine Weyl group symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toda lattice equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B¨ acklund transformations and discrete dPI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coalescence from PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 6 10 12 15 18 18 21 22 23 24 25
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P.J. Forrester, N.S. Witte
˜N (s; a) . . . 4.1 Calculation of EN (0; (s, ∞)) and E 4.2 Calculation of FN (s; a) . . . . . . . . . . . . . . 4.3 UN (t; a) and VN (t; a) as Painlev´ e transcendents 5. Edge Scaling in the GUE . . . . . . . . . . . . . . . ˜ soft (s; a) . . . . . 5.1 Calculation of E soft (s) and E 5.2 Calculation of F soft (λ; a) . . . . . . . . . . . . . 6. Conclusions - A Programme . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .
3. τ -Function Theory for PII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. Application to Finite GUE Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction and Summary Hermitian random matrices X with a unitary symmetry are defined so that the joint distribution of the independent elements P (X ) is unchanged by the similarity transformation X → U † XU for U unitary. For example, an N ∞ ensemble of matrices with P (X ) := exp j =0 αj Tr(X j ) =: j =1 g (λj ) for general g (x) ≥ 0 possesses a unitary symmetry. Such ensembles have the property that the corresponding eigenvalue probability density function p(λ1 , . . . , λN ) is given by the explicit functional form 1 p(λ1 , . . . , λN ) = C
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P.J. Forrester1 , N.S. Witte1,2
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Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia. E-mail: P.Forrester@.au E-mail: N.Witte@.au, School of Physics, University of Melbourne, Victoria 3010, Australia.
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Communications in Mathematical Physics manuscript No. (will be inserted by the editor)