[工程科技]Limit Correlation Functions for Fixed Trace Random Matrix Ensembles.pdfVIP

LIMIT CORRELATION FUNCTIONS 7 FOR FIXED TRACE 0 RANDOM MATRIX ENSEMBLES 0 2 ¨ 1 1,2 c FRIEDRICH GOTZE AND MIKHAIL GORDIN e D 2 1 Abstract. Universal limits for the eigenvalue correlation func- tions in the bulk of the spectrum are shown for a class of non- ] R determinantal random matrices known as the fixed trace or the P Hilbert-Schmidt ensemble. These universal limits have been proved . before for determinantal Hermitian matrix ensembles and for some h special classes of the Wigner random matrices. t a m 1. Introduction and the statement of the result [ 2 Let HN be the set of all N ×N (complex) Hermitian matrices, and let v tr A = N a denote the trace of a square matrix A = (a )N . H 9 i=1 ii ij i,j =1 N 4 is a real Hilbert space of dimension N2 with respect to the symmetric 1 bilinear form (A, B) → tr AB. Let lN denote the unique Lebesgue 0 1 measure on HN which satisfies the relation lN (Q) = 1 for every cube 6 Q ⊂ HN with edges of length 1. A Gaussian probability measure on 0 HN invariant with respect to all orthogonal linear transformations of / h HN is uniquely defined up to a scaling transformation. Such measures t a form a one-parameter family (s )s0 , where the measure s is specified