A random matrix decimation procedure relating β=2/(r+1) to β=2(r+1)
aa r X i v : . [ m a t h - ph ] N ov A random matrix decimation procedure relating β = 2 / ( r + 1) to β = 2( r + 1) Peter J. Forrester † Department of Mathematics and Statistics, University of Melbourne,Victoria 3010, Australia
Abstract
Classical random matrix ensembles with orthogonal symmetry have the property that the jointdistribution of every second eigenvalue is equal to that of a classical random matrix ensemblewith symplectic symmetry. These results are shown to be the case r = 1 of a family of inter-relations between eigenvalue probability density functions for generalizations of the classicalrandom matrix ensembles referred to as β -ensembles. The inter-relations give that the jointdistribution of every ( r + 1)-st eigenvalue in certain β -ensembles with β = 2 / ( r + 1) is equalto that of another β -ensemble with β = 2( r + 1). The proof requires generalizing a conditionalprobability density function due to Dixon and Anderson.1 Introduction
The Dixon-Anderson conditional probability density function (PDF) refers to the function of { λ j } specified by [3, 1]Γ( P nj =1 s j )Γ( s ) · · · Γ( s n ) Q ≤ j Let r ∈ Z + . The Dixon-Anderson PDF (1.1) is the r = 1 case of the family ofconditional PDFs C Q ≤ j 4n (1.12) and b = 0 in (1.14). In the study of these ground states, identities of a different typeto Theorem 2 relating β to 4 /β ( β even) have previously been encountered. These are so calledduality relations, an example being [2] D N Y j =1 ( t − x j ) m E ME β,N ( e − βx / ) = D m Y j =1 ( t − ix j ) N E ME /β,m ( e − x ) . The four examples of (1.12) or (1.14) which permit interpretations as the ground state ofquantum many body systems can also be realized as the eigenvalue PDF for certain ensemblesof random matrices. Thus, for example, with ˜ χ k denoting value drawn from the square rootof the gamma distribution Γ[ k/ , , 1] denoting a number drawn from the standardnormal distribution, the tridiagonal matrix T β := N[0 , 1] ˜ χ ( N − β ˜ χ ( N − β N[0 , 1] ˜ χ ( N − β ˜ χ ( N − β N[0 , 1] ˜ χ ( N − β . . . . . . . . .˜ χ β N[0 , 1] ˜ χ β ˜ χ β N[0 , has its eigenvalue PDF given by ME β,N ( e − x / ) [4]. Taking various scaled N → ∞ limits(see Section 5.2) of these ensembles of random matrices leads to a description of the limitingeigenvalue distributions in terms of stochastic differential operators [7, 19, 16]. Further, foreigenvalues in the bulk, there is a description in terms of a process associated with Brownianmotion in the hyperbolic plane [21].It is an open problem, for general r , to derive the results of Theorem 2 as consequences ofthe eigenvalue PDFs being realizable from concrete random matrix ensembles. In the case r = 1such matrix theoretic derivations have been given (excluding the sixth identity which relates toCE bN ) using matrix realizations not applicable for general r [12, 13]. L r,n ( { a p } ) — a cancellation effect Let L r,n ( { a p } ) := Z A r dλ · · · dλ r ( n − Y ≤ j Consider an arrangement of r ′ s and q ′ s (1 ≤ q ≤ r ) in a line. Let this beconsidered as the sequence A = ( n j ) j =1 ,...,r + q with each n j = 0 or 1. Further let K ( n j ) = (cid:26) , n j = 0 ′ s to the right of n j , n j = 1 , (2.4) and use this to specify the statistic K ( A ) = r + q X j =1 K ( n j ) . (2.5) One has X A e − πiK ( A ) / ( r +1) = 0 . (2.6)Proof. The definition (2.4) can be written K ( n j ) = n j r + q X k = j +1 (1 − n j ) , and this substituted in (2.5) gives K ( A ) = r + q X j =1 n j ( r + q − j ) − r + q X j 1. This analytic continuation corresponds to the limit that thecontours tend to the real line in the second configuration of Figure 1, and the endpoints { ˜ a j } are appropriately related to the endpoints { a j } . real axisreal axisa l−1 aa a a l+1 l l−1 a l+2 a l+2 ~ ~ ~ a ~ l l+1 Figure 1: The contours from a l +2 to a l +1 , a l +1 to a l , and a l to a l − are deformed to the contoursjoining the corresponding tilded variables. Our interest is in the limit that ˜ a j = a j ( j = l, l + 1),˜ a l = a l +1 , ˜ a l +1 = a l and all contours in the second diagram run along the real axis. In the case l = n − a l +2 to ˜ a l +1 is to be deleted, while in the case l = 1 the contour from˜ a l to ˜ a l − is to be deleted.To study the integrand of (2.1) in the case of the second configuration, for notational con-venience set λ ( j − r + ν = λ ( ν ) j ( ν = 1 , . . . , r ), which are to be referred to as species j . Then forthe first configuration a j +1 < λ ( r ) j < λ ( r − j < · · · < λ (1) j < a j ( j = 1 , . . . , n − , (cid:16) n Y j =1 Y ≤ ν ≤ µ ≤ r ( λ ( ν ) j − λ ( µ ) j ) / ( r +1) (cid:17)(cid:16) Y ≤ j 1, and suppose that to begin the r species l variables are to theleft of the q species l − a l , ˜ a l +1 ). We see from (2.9) that interchangingthe position of coordinates corresponding to different species does not change the magnitude ofthe integrand but it does change the phase, with each interchange of a species l − l contributing e − πi/ ( r +1) . Hence for a general ordering of the r species l variables and q species l − r + q ) positions in (˜ a l , ˜ a l +1 ) thephase is given by e − πiK ( A ) / ( r +1) . (2.13)Here K ( A ) is as in Lemma 1 with the 0’s corresponding to species l and the 1’s to species l − q = 0, p ≥ 1. It remains to consider the cases p, q ≥ 1. In such cases, with the positionsof the species l − l + 1coordinates), we see that the contribution to the phase of each such coordinate is equal to e − πil ∗ / ( r +1) , where l ∗ is the number of both species l , l + 1 to its left, and in particular isindependent of their ordering. But we know that summing over this latter ordering gives thecancellation (2.1), so in all cases there is no contribution from non-empty configurations (2.11)and (2.12). 8s a consequence of both (2.11) and (2.12) having to be empty for a non-zero contributionto the contour integral, it follows that (2.10) can be supplemented by the requirements that˜ a l > ˜ λ (1) l +1 > ˜ λ (2) l +1 > · · · ˜ λ ( r ) l +1 > ˜ a l +2 ˜ a l − > ˜ λ (1) l − > ˜ λ (2) l − > · · · ˜ λ ( r ) l − > ˜ a l +1 . Up to a phase, this contour integral is precisely (2.1) with the position of a l and a l +1 in-terchanged, and correspondingly s l and s l +1 interchanged. The phase is straightforward tocalculate, giving as the final result (2.3). (cid:3) As remarked below (2.2), we must show that L r,n ( { a p } ) is proportional to R r,n ( { a p } ), and thendetermine the proportionality. For the former task, our strategy is to show that L r,n ( { a p } )factorizes into a term singular in { a p } , and a term analytic in { a p } . The singular factor isprecisely R r,n ( { a p } ), while a scaling argument shows that the analytic factor must be a constant.Intermediate working relating to the singular terms allows the proportionality to be determined.Consider L r,n ( { a p } ) as an analytic function of a in the appropriately cut complex a -plane.Singularites occur as a approach any of a , . . . , a n . The singular behaviour as a approaches a can be determined directly from (2.1). Thus, as a → a the integral over species 1 effectivelyfactorizes from the integral over the other species, showing L r,n ( { a p } ) = n Y p =3 ( a − a p ) r ( s p − I r ( a , a ) × L r,n − ( { a p } p =2 ,...,n ) | s s + s +2 / ( r +1) − F ( a − a ; { a p } p =2 ,...,n ) (2.14)where F ( z ; { a p } p =2 ,...,n ) is analytic about z = 0 and equal to unity at z = 0, and I r ( a , a ) := Z a >λ (1)1 > ··· >λ ( r )1 >a dλ (1)1 · · · dλ ( r )1 × Y ≤ ν ≤ µ ≤ r ( λ ( ν )1 − λ ( µ )1 ) / ( r +1) r Y ν =1 ( a − λ ( ν )1 ) s − ( λ ( ν )1 − a ) s − . (2.15)Thus the singular behaviour is determined by the singular behaviour of I r ( a , a ). This in turnis revealed by a simple scaling of the integrand, which shows I r ( a , a ) = ( a − a ) r ( r − / ( r +1)+ r ( s + s − r ! S r ( s − , s , / ( r + 1)) (2.16)where S n ( λ , λ , λ ) denotes the Selberg integral (1.4).For the singular behaviour as a approaches a k ( k = 2), we make use of Proposition 1which says that up to a phase the function of { a p } p =1 ,...,n obtained from L r,n ( { a p } ) by analyticcontinuation is symmetric in { ( a p , s p ) } . Hence as a function of a it must be that L r,n ( { a p } ) = n Y k =2 ( a j − a k ) r ( r − / ( r +1)+ r ( s j + s k − ˜ F ( a ; { a p } p =2 ,...,n ) , (2.17)where ˜ F is analytic in a . Further, repeating the argument with L r,n ( { a p } ) regarded as afunction of a , . . . , a n in turn shows L r,n ( { a p } ) = R r,n ( { a p } ) G ( { a p } ) (2.18)9here G is analytic in { a p } and symmetric in { ( a p , s p ) } .It remains to determine G . This can be done by considering the scaling properties of bothsides of (2.18) upon the replacements { a p } 7→ { ca p } , c > 0. After changing variables λ k cλ k ( k = 1 , . . . , n ( r − L r,n ( { ca p } ) = c r ( n − r ( n − r ( n − − / ( r +1)+ r ( n − P np =1 ( s p − L r,n ( { a p } ) (2.19)while we read off from (2.2) that R r,n ( { ca p } ) = (cid:16) Y ≤ j 1, and recalling(3.25). It has been revised in Section 3 that the Dixon-Anderson density (1.1) with s j = 1 impliesthe decimation identities for superimposed ensembles (3.10), (3.11). These results rely on theeigenvalue PDF of OE n ( f ) ∪ OE n +1 ( f ) being proportional to (3.12) and OE n ( f ) ∪ OE n ( f ) beingproportional to (3.13). No generalization of these latter facts for ensembles of the form (1.12) isknown. Instead let us make note of a further interpretation of the Dixon-Anderson integral inthe case s j = 1, or more explicitly in the case of the parameters (3.4) with the final conditionreplaced by s j = 1 ( j = 2 , . . . , N + 1), which does permit a generalization.In the latter circumstance, one has that Z A dλ · · · dλ N +1 N +1 Y j =1 λ aj (1 − λ j ) b Y ≤ j 0, this corresponds to the PDF forthe ( k + 1)-st eigenvalue as labelled from the smallest. We then read off from the fourth, thirdand sixth relations in (1.16) the following result (analogous results hold for the other relations;however these three are representative of all situations). Corollary 1. One has p max (( r + 1) k + r ; s ; ME / ( r +1) , ( r +1) N + r ( e − x )) = p max ( k ; s ; ME r +1) ,N ( e − ( r +1) x )) p min (( r + 1) k + r ; s ; ME / ( r +1) , ( r +1) N + r ( x a e − x )) = p min ( k ; s ; ME r +1) ,N ( x ( r +1) a +2 r e − ( r +1) x )) p min ((( r + 1) k + r ; s ; CE b / ( r +1) , ( r +1) N + r = p min ( k ; s ; CE ( r +1) b +2 r r +1) ,N ) . (5.3)Recalling (3.25) we see that with b = 2 / ( r + 1) the final equation in (5.3) implies p spacing (( r + 1) k + r ; s ; CE / ( r +1) , ( r +1) N ) = p spacing ( k ; s ; CE r +1) ,N ) , (5.4)where p spacing ( k ; s ; CE β,N ) is the PDF for the spacing between eigenvalues which are ( k + 1)-stneighbours.The three situations that the results of Corollary 1 are representative of the soft edge, thehard edge, and a spectrum singularity in the bulk. The soft edge is the neighbourhood of thelargest eigenvalue, so called because the eigenvalue density has support beyond this region. Thehard edge is the neighbourhood of the smallest eigenvalue, in the situation that the eigenvaluesupport is strictly zero for x < 0. The bulk is the portion of the spectrum a macroscopic distancefrom the edges (all circular ensembles satisfy this requirement), while a spectrum singularitycorresponds to a factor of the form | x | α (for x → 0) in the one body weight.16ach permits a scaling in which the origin is shifted to the appropriate neighbourhood (thismust be done in the case of the soft edge only), and the eigenvalues are scaled so that the spacingbetween eigenvalues is of order unity. For the N → ∞ limit of the ensemble ME β,N ( e − cx ), thesoft edge scaling is given by [10] x β c (4 N + 2(2 N ) / s β x ) , (5.5)where s β > k + 1)-st largest eigenvalue is thengiven by lim N →∞ βc (2 N ) / s β p max ( k ; s ; ME β,N ( e − cx )) =: p soft β ( k ; s ) . (5.6)The hard edge scaling of ME β,N ( x a e − cx ) is given by x β c x N ˜ s β (5.7)where ˜ s β > k + 1)-st smallest eigenvalue atthe hard edge is thuslim N →∞ β c N ˜ s β p min ( k ; s ; ME β,N ( x a e − cx )) =: p hard β ( k ; s ; a ) . (5.8)Finally, in the ensemble CE bβ,N the mean spacing between eigenvalues is 2 π/N , and we havelim N →∞ πN p min ( k ; 2 πs/N ; CE bβ,N ) =: p bulk , s . s .β ( k ; s ; b ) (5.9)for the PDF of the ( k + 1)-st eigenvalue to the right of a spectrum singularity in the bulk withunit density. Taking these limits in the results (5.3) gives inter-relations between the scaledPDFs with β = 2 / ( r + 1) and β = 2( r + 1). Proposition 2. Choose the scale in (5.5) such that s / ( r +1) = ( r + 1) / s r +1) , and choose thescale in (5.7) such that ˜ s / ( r +1) ( r + 1) = ˜ s r +1) . One has p soft2 / ( r +1) (( r + 1) k + r ; s ) = p soft2( r +1) ( k ; s ) p hard2 / ( r +1) (( r + 1) k + r ; s ; a ) = p hard2( r +1) ( k ; s ; ( r + 1) a + 2 r )( r + 1) p bulk , s . s . / ( r +1) (( r + 1) k + r ; ( r + 1) s ; b ) = p bulk , s . s . r +1) ( k ; s ; ( r + 1) b + 2 r ) . (5.10)Let p bulk , sp .β ( k ; s ) denote the PDF for a spacing of size s between eigenvalues which are ( k +1)-st neighbours, in the bulk of the circular ensemble CE β,N scaled so that the eigenvalue densityis unity in the N → ∞ limit. This corresponds to the third relation in (5.10) with b = 2 / ( r + 1)(recall the discussion leading to (5.4)) and so we have( r + 1) p bulk , sp . / ( r +1) (( r + 1) k + r ; ( r + 1) s ) = p bulk , sp . r +1) ( k ; s ) . (5.11)As a consistency check on (5.11) it can be shown to be compatible with the asymptotic form(1.17). First one recalls that in general p bulk , sp .β ( k ; s ) is related to { E bulk β ( n ; s ) } n =0 , ,...,k accordingto the formula (see e.g. [9, Ch. 6]) p bulk , sp .β ( k ; s ) = d ds k X j =0 ( k − j + 1) E bulk β ( j ; s ) . s the term j = k will dominate, and so (5.11) requires that for large s ( r + 1) E bulk2 / ( r +1) (( r + 1) k ; ( r + 1) s ) ∼ E bulk2( r +1) ( k ; s ) . Taking logarithms of both sides and comparing with (1.17) gives precise agreement with thelatter. Acknowledgements The idea of seeking inter-relations of the type reported on here is due to B´alint Vir´ag, commu-nicated to the author at the AMS-IMS-SIAM summer research conference on Random MatrixTheory, Integrable Systems, and Stochastic Processes (June, 2007). Ths work has been sup-ported by the Australian Research Council. References [1] G.W. Anderson. A short proof of Selberg’s generalized beta formula. Forum Math. , 3:415–417, 1991.[2] T.H. Baker and P.J. Forrester. 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