STATPHYS 20, Paris, 20–24 juillet 1998, T1028:PO01/50 A problem of random matrices: hermitian matrices coupled in a chain correlations and spacings B. Eynard, G. Mahoux, M.L. Mehta and J.-M. Normand CEA/Saclay, Service de Physique Th´ eorique F-91191 Gif-sur-Yvette Cedex, FRANCE I. Description of the model The configurations of the statistical system considered here are described by one or several n × n matrices A j . The statistical weight of a particular configuration is assumed to be the exponential of some energy function, exp( − H ). The energy H contains two kinds of terms ∑ ∑ H = tr V j ( A j ) − c j tr ( A j A j +1 ) j j one matrix terms V j ( A ) and couplings between two matrices. The only tractable problem is that of a linear open chain of matrices with couplings between adjacent matrices, as in the above equation. Three types of matrices are usually investigated: real symmetric, complex hermitian and quaternion self-dual. Such theoretical models are supposedly relevant to describe the physics of nuclear excitations, conductance of heterogeneous metallic wires, and even quantum gravity. The physical questions to be asked, and hopefully answered, are: 1. What is the distribution of the eigenvalues of one or several matrices ? 2. What are the correlations between the eigenvalues of the same or different matrices ? 3. What is the probability of finding a given number of eigenvalues in given intervals ? ... From the mathematical point of view the case of complex hermitian matrices is the simplest and the above three questions can be almost fully answered, while the most relevant for physical applications is the case of real symmetric matrices. A common feature of all our results is that the quantities in the above questions (prob- ability density of the eigenvalues, correlation functions and generating function of the spac- ings) are expressed in terms of a single determinant. For clarity we present below separately the cases of one matrix and of several coupled matrices. Only the coupling of matrices in an open chain is considered. 1
II. One n × n hermitian matrix (summary of known results) Probability density of the matrix elements F ( A ) = C exp[ − tr V ( A )] the function V ( A ) depending only on the eigenvalues of A . Here and in what follows C is a generic symbol for a constant. Eigenvalues of the matrix A : x = { x 1 , ..., x n } . 1. Probability density of the eigenvalues [ n ] F ( x ) = C ∆( x ) 2 exp ∑ − V ( x i ) i =1 ∏ ∆( x ) = ( x i − x j ) 1 ≤ i<j ≤ n 2. Eigenvalue correlations Let P r ( ξ ) be a monic polynomial (i.e. the coefficient of its highest power is one) of order r in ξ , chosen such that ∫ P r ( ξ ) P s ( ξ ) exp[ − V ( ξ )] dξ = h r δ rs and let ] n − 1 [ − 1 2 V ( ξ ) − 1 1 ∑ K ( ξ, η ) = exp 2 V ( η ) P r ( ξ ) P r ( η ) . h r r =0 Then the density of an ordered set of k eigenvalues of A within small intervals around x 1 , ..., x k is n n ! ∫ ∏ R k ( x 1 , ..., x k ) = F ( x ) dx r = det[ K ( x r , x s )] r,s =1 ,...,k . ( n − k )! r = k +1 For a real symmetric or a quaternion self-dual matrix A , similar formulae are known. 2
3. Spacing functions The spacing function E ( k, I ), i.e. the probability that a chosen domain I contains exactly k eigenvalues of A , (0 ≤ k ≤ n ), is k n n ! ∫ ∏ ∏ dx 1 ...dx n E ( k, I ) = F ( x ) χ ( x j ) [1 − χ ( x j )] k !( n − k )! j =1 j = k +1 ( d � ) k = 1 � R ( z, I ) � k ! dz � � z = − 1 where χ ( x ) is the characteristic function of the domain I , χ ( x ) = 1, if x ∈ I and χ ( x ) = 0, otherwise, and R ( z, I ) is the generating function of the integrals over I of the correlation functions R k ( x 1 , ..., x k ), n n ∫ ρ k ∏ ∑ k ! z k R ( z, I ) = F ( x ) [1 + zχ ( x j )] dx j = 1 + j =1 k =1 k ∫ ∏ ρ k = R k ( x 1 , ..., x k ) χ ( x j ) dx j . j =1 R ( z, I ) can be expressed as a n × n determinant R ( z, I ) = det[ G ij ] i,j =0 ,...,n − 1 G ij = 1 ∫ P i ( x ) P j ( x ) exp[ − V ( x )][1 + zχ ( x )] dx = δ ij + ¯ G ij h i or as the Fredholm determinant n ∏ R ( z, I ) = [1 + λ k ( z, I )] k =1 of the integral equation ∫ N ( x, y ) f ( y ) dy = λf ( x ) where remarkably the kernel N ( x, y ) is simply zK ( x, y ) χ ( y ) with the same K ( x, y ) ap- pearing in the correlation functions. The λ i ( z, I ) are the eigenvalues of the above equation and also of the matrix [ ¯ G ij ]. Similar expressions of E ( k, I ) for real symmetric or quaternion self-dual matrices are known. 3
III. Case of p n × n matrices coupled in an open chain Probability density of the matrix elements [ { 1 2 V 1 ( A 1 ) + V 2 ( A 2 ) + · · · + V p − 1 ( A p − 1 ) + 1 }] F ( A 1 , · · · , A p ) = C exp − tr 2 V p ( A p ) × exp [tr ( c 1 A 1 A 2 + c 2 A 2 A 3 + · · · + c p − 1 A p − 1 A p )] where the V j ( A ) are functions depending only on the eigenvales of A and the c j are real constants. Eigenvalues of the matrix A j : x j = { x j 1 , ..., x jn } , 1 ≤ j ≤ p . 1. Probability density of the eigenvalues [ n }] { 1 2 V 1 ( x 1 r ) + V 2 ( x 2 r ) + · · · + V p − 1 ( x p − 1 r ) + 1 ∑ F ( x 1 ; ... ; x p ) = C exp − 2 V p ( x pr ) r =1 × ∆( x 1 )∆( x p ) × det [ e c 1 x 1 r x 2 s ] det [ e c 2 x 2 r x 3 s ] · · · det [ e c p − 1 x p − 1 r x ps ] [ p − 1 ] ∏ = C ∆( x 1 )∆( x p ) det [ w k ( x kr , x k +1 s )] r,s =1 ,...n k =1 [ − 1 2 V k ( ξ ) − 1 ] where ∆( x j ) = ∏ 1 ≤ r<s ≤ n ( x jr − x js ) and w k ( ξ, η ) = exp 2 V k +1 ( η ) + c k ξη . For real symmetric or quaternion self-dual matrices A j similar formulae are not known. 4
2. Eigenvalue correlations (new results) Let P r ( ξ ) = P 1 r ( ξ ) and Q r ( ξ ) = Q pr ( ξ ) be monic polynomials of order r in ξ , chosen such that ∫ P r ( ξ 1 ) w 1 ( ξ 1 , ξ 2 ) w 2 ( ξ 2 , ξ 3 ) ...w p − 1 ( ξ p − 1 , ξ p ) Q s ( ξ p ) dξ 1 ...dξ p = h r δ rs and let ∫ P jr ( ξ ) = P r ( ξ 1 ) w 1 ( ξ 1 , ξ 2 ) w 2 ( ξ 2 , ξ 3 ) ...w j − 1 ( ξ j − 1 , ξ ) dξ 1 ...dξ j − 1 , 1 < j ≤ p ∫ Q jr ( ξ ) = w j ( ξ, ξ j +1 ) w j +1 ( ξ j +1 , ξ j +2 ) ...w p − 1 ( ξ p − 1 , ξ p ) Q r ( ξ p ) dξ j +1 ...dξ p , 1 ≤ j < p n − 1 1 ∑ H ij ( ξ, η ) = Q ir ( ξ ) P jr ( η ) , 1 ≤ i, j ≤ p h r r =0 { ∫ w i ( ξ, ξ i +1 ) w i +1 ( ξ i +1 , ξ i +2 ) ...w j − 1 ( ξ j − 1 , η ) dξ i +1 ...dξ j − 1 , 1 ≤ i < j ≤ p E ij ( ξ, η ) = 0 , i ≥ j K ij ( ξ, η ) = H ij ( ξ, η ) − E ij ( ξ, η ) . Then the density of ordered sets of k j eigenvalues of A j in small intervals around x j 1 , ..., x jk j for j = 1, ..., p is a m × m determinant with m = k 1 + ... + k p p n ∫ n ! ∏ ∏ = det [ K ij ( x ir , x js )] i, j = 1 , ..., p F ( x 1 ; ... ; x p ) dx jr j . ( n − k j )! j =1 r j = k j +1 r = 1 , ..., k i s = 1 , ..., k j For real symmetric or quaternion self-dual matrices A j similar formulae are not known. 5
3. Spacing functions (new results) The spacing function E ( k 1 , I 1 ; ... ; k p , I p ), i.e. the probability that the domain I j contains exactly k j eigenvalues of the matrix A j for j = 1 , ..., p , 0 ≤ k j ≤ n , (the domains I j may have overlaps), is as in the one matrix case ( ∂ ( ∂ � ) k 1 ) k p 1 ... 1 � E ( k 1 , I 1 ; ... ; k p , I p ) = R ( z 1 , I 1 ; ... ; z p , I p ) � k 1 ! k p ! ∂z 1 ∂z p � � z 1 = ... = z p = − 1 with the generating function p n ∫ ∏ ∏ R ( z 1 , I 1 ; ... ; z p , I p ) = F ( x 1 ; ... ; x p ) [1 + z j χ j ( x jr )] dx jr j =1 r =1 χ j ( x ) being the characteristic function of the domain I j . The generating function R ( z 1 , I 1 ; ... ; z p , I p ) can again be expressed as a n × n determinant R ( z 1 , I 1 ; ... ; z p , I p ) = det[ G ij ] i,j =0 ,...,n − 1 where [ p [ p − 1 ] ] G ij = 1 ∫ ∏ ∏ = δ ij + ¯ P i ( x 1 ) w k ( x k , x k +1 ) Q j ( x p ) { 1 + z k χ k ( x k ) } dx k G ij h i k =1 k =1 or as the Fredholm determinant n ∏ R ( z 1 , I 1 ; ... ; z p , I p ) = [1 + λ k ( z 1 , I 1 ; ... ; z p , I p )] k =1 of an integral equation ∫ N ( x, x p ) f ( x p ) dx p = λf ( x ) . Here the kernel N ( x, y ) is more complicated than in the one matrix case ] [ p n − 1 [ p − 1 ] [ p − 1 ] 1 ∫ ∑ ∏ ∏ ∏ N ( x, x p ) = Q ℓ ( x ) P ℓ ( x 1 ) w k ( x k , x k +1 ) { 1 + z k χ k ( x k ) } − 1 dx k . h ℓ ℓ =0 k =1 k =1 k =1 For real symmetric or quaternion self-dual matrices A j nothing similar is known. 6
E-mail addresses of the authors B. Eynard: eynard@spht.saclay.cea.fr G. Mahoux: mahoux@spht.saclay.cea.fr M.L. Mehta: mehta@spht.saclay.cea.fr J.-M. Normand: norjm@spht.saclay.cea.fr References B. Eynard and M.L. Mehta, Matrices coupled in a chain. I. Eigenvalue correlations, J. Phys. A: Math. Gen. 31 (1998) 4449–4456. G. Mahoux, M.L. Mehta and J.-M. Normand, Matrices coupled in a chain. II. Spacing functions, J. Phys. A: Math. Gen. 31 (1998) 4457–4464. 7
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