On the mean Density of States of some matrices related to the beta ensembles and an application to the Toda lattice
OOn the mean Density of States of some matrices related tothe beta ensembles and an application to the Toda lattice
G. Mazzuca ∗ September 17, 2020
Abstract
In this manuscript we study tridiagonal random matrix models related to the classical β -ensembles (Gaussian, Laguerre, Jacobi) in the high temperature regime, i.e. when thesize N of the matrix tends to infinity with the constraint that βN “ α constant, α ą .We call these ensembles the Gaussian, Laguerre and Jacobi α -ensembles and we prove theconvergence of their empirical spectral distributions to their mean densities of states andwe compute them explicitly. As an application we explicitly compute the mean density ofstates of the Lax matrix of the Toda lattice with periodic boundary conditions with respectto the Gibbs ensemble. In this manuscript we consider some tridiagonal random matrix models related to the classical β -ensembles [7,10,15]. More specifically we study the mean density of states of the random matricesin Table 1 where the quantity N p , σ q is the real Gaussian random variable with density e ´ x σ ? πσ supported on all R , the quantity χ α is the chi-distribution with density x α ´ e ´ x α ´ Γ p α q supported on R ` , here Γ p α q is the gamma function, and Beta p a, b q is the Beta random variable with density Γ p a ` b q x a ´ p ´ x q b ´ Γ p a q Γ p b q supported on p , q .Let us explain some terminology first and then state our result.A random Jacobi matrix is a symmetric tridiagonal N ˆ N matrix of the form T N : “ ¨˚˚˚˚˚˚˝ a b b a b . . . . . . . . .. . . . . . b N ´ b N ´ a N ˛‹‹‹‹‹‹‚ (1.1)where t a i u Ni “ are i.i.d. real random variables and t b i u N ´ i “ are i.i.d. positive random variablesindependent from the a i . This matrix has the property of having N -distinct eigenvalues [6]. The ∗ International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
Email: [email protected] a r X i v : . [ m a t h . SP ] S e p aussian α Ensemble H α „ ? ¨˚˚˚˝ a b b a b . . . . . . . . . b N ´ a N ˛‹‹‹‚ ,H α P Mat p N ˆ N q ,b n „ χ α n “ , . . . , N ´ ,a n „ N p , q n “ , . . . , N , Laguerre α Ensemble L α,γ “ B α,γ B (cid:124) α,γ , B α,γ “ ? ¨˚˚˚˝ x y x . . . . . . y N ´ x N ˛‹‹‹‚ ,B α,γ P Mat p N ˆ M q , M ě N,x n „ χ αγ n “ , . . . , N,y n „ χ α n “ , . . . , N ´ , Jacobi α Ensemble J α “ D α D (cid:124) α , D α “ ¨˚˚˚˝ s t s . . . . . . t N ´ s N ˛‹‹‹‚ ,D α P Mat p N ˆ N q ,t n “ a q n p ´ p n q , s n “ a p n p ´ q n ´ q ,q n „ Beta p α, α ` a ` b ` q p q “ q ,p n „ Beta p α ` a ` , α ` b ` q . Table 1: The Gaussian, Laguerre and Jacobi α -ensembles.empirical spectral distribution of T N is the random probability distribution on R defined as d ν p N q T : “ N N ÿ j “ δ λ p N q j , (1.2)where λ p N q ą . . . ą λ p N q N are the eigenvalues of T N and δ p¨q is the delta function. The mean Density of States d ν T is the non random probability distribution, provided it exists, defined as @ d ν T , f D : “ lim N Ñ8 E ”A d ν p N q T , f Eı , (1.3)for all continuous and bounded functions f , here x d σ, f y : “ ş R f d σ and E r¨s stands for theexpectation with respect to the given probability distribution.In this manuscript we identify the mean density of states of the Gaussian, Laguerre andJacobi α -ensembles introduced in Table 1. Theorem 1.1.
Consider the matrices H α , L α,γ , and J α in Table 1 with α ě , γ P p , q , a ` α ą , b ` α ą and a R N . Then their empirical spectral distributions d ν p N q H , d ν p N q L , and d ν p N q J converge almost surely, in the large N limit, to their corresponding mean density of states,whose formula are given explicitly by: ν H p x q “ B α p αµ α p x qq d x , (1.4) d ν L p x q “ B α p αµ α,γ p x qq d x , x ě , (1.5) d ν J p x q “ B α p αµ α,a,b p x qq d x , ď x ď . (1.6) Here B α is the derivative with respect to α and µ α p x q : “ e ´ x ? π ˇˇˇ p f α p x q ˇˇˇ ´ , p f α p x q : “ c α Γ p α q ż t α ´ e ´ t e ixt d t , (1.7) µ α,γ p x q : “ p α ` q Γ ´ ` αγ ` α ¯ x αγ e ´ x (cid:12)(cid:12)(cid:12) ψ ´ α, ´ αγ ; xe ´ iπ ¯ (cid:12)(cid:12)(cid:12) x ě , (1.8) with Γ p z q the gamma-function and ψ p v, w ; z q is the Tricomi’s confluent hypergeometric function,for the definition see Appendix A, and µ α,a,b p x q : “ Γ p α ` q Γ p α ` a ` b ` q Γ p α ` a ` q Γ p α ` b ` q x a p ´ x q b (cid:12)(cid:12)(cid:12) U p x q ` e iπb V p x q (cid:12)(cid:12)(cid:12) ď x ď , (1.9) where U p x q : “ Γ p α ` q Γ p a ` q Γ p ` α ` a q F p α, ´ α ´ a ´ b ´ , ´ a ; x q , (1.10) V p x q : “ ´ πα Γ p α ` a ` b ` q sin p πa q Γ p ` α ` b q Γ p a ` q p ´ x q b ` x a ` F p ´ α, α ` a ` b ` , ` a ; x q , (1.11) here F p a, b, c ; z q is the Hypergeometric function: F p a, b, c ; z q : “ ÿ n “ p a q n p b q n p c q n z n n ! , p a q n : “ a p a ` q ¨ ¨ ¨ p a ` n ´ q . (1.12) Moreover, for any non trivial polynomial P p x q the following limits hold: ? N ´A d ν p N q H , P p x q E ´ @ d ν H , P p x q D¯ d Ñ N p , σ P q as N Ñ 8 , (1.13) ? N ´A d ν p N q L , P p x q E ´ @ d ν L , P p x q D¯ d Ñ N p , σ P q as N Ñ 8 , (1.14) ? N ´A d ν p N q J , P p x q E ´ @ d ν J , P p x q D¯ d Ñ N p , r σ P q as N Ñ 8 , (1.15) for some constants σ P , σ P , r σ P ě , here d Ñ is the convergence in distribution. In figures 1–3 we plot the empirical spectral distribution of the α -ensembles for differentvalues of the parameters.The measures with density µ α , µ α,γ and µ α,a,b have already appeared in the literature asthe orthogonality measures of the associated Hermite, Laguerre and Jacobi polynomias ( seeAppendix A). Such measures have also appeared in the study of the classical β -ensembles [7](see Table 2) in the high temperature regime , namely in the limit when N Ñ 8 , with βN Ñ α ,3 ą , [5, 9, 23, 24]. In order to summarize the results of those papers we recall that for theJacobi matrix T N in (1.1) the spectral measure d µ p N q T is the probability measure supported onits eigenvalues λ p N q , . . . , λ p N q N with weights q , . . . , q N where q j “ |x v p N q j , e y| and v p N q , . . . , v p N q N are the orthonormal eigenvectors: d µ p N q T : “ N ÿ j “ q j δ λ p N q j . (1.16)As the eigenvectors form an orthonormal basis, and || e || “ we get that ř Nj “ q j “ . Moreoverthe set of finite Jacobi matrix of size N is in one to one correspondence with the set of probabilitymeasure supported on N real points [6].For the β -ensembles the quantities t q i u Ni “ are independent from the eigenvalues and aredistributed as p χ β , . . . , χ β q normalized to unit length [7, 10, 15]. It follows that E r q j s “ N .Consequently the mean of the empirical measure (1.2) coincides with the mean of the spectralmeasure (1.16), namely d¯ ν p N q H β “ d¯ µ p N q H β , d¯ ν p N q L β “ d¯ µ p N q L β , d¯ ν p N q J β “ d¯ µ p N q J β , where H β , L β and J β refer to the Hermite, Laguerre and Jacobi β -ensembles. It is shown in [9]( see also [5]) that the measures d¯ ν p N q H β “ d¯ µ p N q H β converge weakly, in the limit N Ñ 8 , with βN “ α , to the non random probability measure with density µ α defined in (1.7). It is shownin [23, 24] that the measures d¯ ν p N q L β “ d¯ µ p N q L β and d¯ ν p N q J β “ d¯ µ p N q J β , under some mild assumptionson the parameters, converge weakly in the limit N Ñ 8 , with βN Ñ α and N { M Ñ γ P p , q to the non random probability measures with density µ α,γ and µ α,a,b defined in (1.8) and (1.9)respectively. In [9, 23, 24] it is showed that these measures coincide with the mean spectralmeasures of the random matrices H α , L α and J α , see Table 1.The problem of convergence of the empirical spectral distribution of the Gaussian, Laguerreand Jacobi α -ensembles has remained unsolved. The present manuscript addresses such problemin Theorem 1.1 by determining the mean Density of States of such random matrices and theirfluctuation. Our strategy to prove the result is the application of the moment method and anastute counting of the super-Motzkin paths [19] to calculate the moments of the the Gaussian,Laguerre and Jacobi α and β -ensembles.For completeness we mention also the result in [20] where a different generalization of theGaussian β ensemble is studied. Indeed in [20] the author examined the mean spectral measureof a random Jacobi matrix T N such that there exists a sequence of real number t m k u k ě and m “ such that E “ p b { N σ q k ‰ Ñ m k as N Ñ 8 for all fixed k P N , which is a generalization ofthe classical case where b „ χ β p N ´ q {? and σ “ { .Finally we relate the Gibbs ensemble of the classical Toda chain to the Gaussian α -ensemble.In particular we obtain, as a corollary of Theorem 1.1, the mean density of states of the Toda Laxmatrix with periodic boundary conditions when the matrix entries are distributed accordingly tothe Gibbs ensemble and when the number of particles goes to infinity. This result is instrumentalto study the Toda lattice in the thermonodynamic limit. We remark that the mean density ofstates of the Toda Lax matrix has already appear in the physics literature [21]. Here we presentan alternative proof of this result. 4aussian β Enseble H β „ ? ¨˚˚˚˚˚˚˝ a b b a b . . . . . . . . .. . . . . . b N ´ b N ´ a N ˛‹‹‹‹‹‹‚ H β P Mat p N ˆ N q ,b n „ χ β p N ´ n q n “ , . . . , N ´ ,a n „ N p , q n “ , . . . , N, Laguerre β Enseble L β,γ “ B β,γ B (cid:124) β,γ , B β,γ “ ? ¨˚˚˚˝ x y x . . . . . . y N ´ x N ˛‹‹‹‚ ,B β,γ P Mat p N ˆ M q , M ě N,x n „ χ β p M ´ n ` q n “ , . . . , N,y n „ χ β p N ´ n q n “ , . . . , N ´ , Jacobi β Enseble J β “ D β D (cid:124) β , D β “ ¨˚˚˚˝ s t s . . . . . . t N ´ s N ˛‹‹‹‚ ,D β P Mat p N ˆ N q ,t n “ a q n p ´ p n q , s n “ a p n p ´ q n ´ q ,q n „ Beta ´ β p N ´ n q , β p N ´ n q ` a ` b ` ¯ p q “ q ,p n „ Beta ´ β p N ´ n q ` a ` , β p N ´ n q ` b ` ¯ . Table 2: The Gaussian, Laguerre and Jacobi β -ensembles In this section we summarize some known results and techniques that we will use along the proofof the main theorem.The moments of a measure d σ , when they exist, are defined as: u p l q : “ @ x l , d σ D l P N . (2.1)Under some mild assumptions, they totally define the measure itself, indeed the following Lemma,whose proof can be found in [4, Lemma B.2], holds: Lemma 2.1. (cf . [4, Lemma B.2]) Let t u p l q u l ě be the sequence of moments of a measure d σ .If lim l Ñ8 inf p u p l q q l l ă 8 , (2.2) then d σ is uniquely determined by the moment sequence t u p l q u l ě . This implies that if two measures have the same moment sequence and (2.2) holds then thetwo measure are the same. We will exploit this property, indeed we will show that the momentsof the random matrices H α , L α and J α coincide, in the large N limit, with the moments of the5easure d ν H p x q , d ν L p x q and d ν J p x q in (1.4)–(1.6) and we will prove that (2.2) holds for all ofthem. This technique undergoes the name of moment method.In order to apply this idea, we need to compute explicitly the moments of the mean densityof states for the α and β -ensembles. We will use the following identity for the moments of themean density of states: @ d ν T , x l D “ lim N Ñ8 N E “ Tr p T lN q ‰ , (2.3)where Tr p T lN q : “ N ÿ j “ T lN p j, j q , and T lN p j, i q is the entry p j, i q of the matrix T lN and the average is made according to thedistribution of the matrix entries. From now on we will write E r f p a , b qs T as the mean valueof f p a , b q made according to the distributions of the matrix T ’s entries, here a is a vector ofcomponents a , . . . , a N .To conclude the computation of the moments, we need an explicit expression for the terms T lN p j, j q . The following lemma proved in [12] gives us their general expressions: Theorem 2.2. (cf. [12, Theorem 3.1]) For any ď l ă N , consider the tridiagonal matrix T N (1.1) , then one has Tr p T lN q “ N ÿ j “ h p l q j , (2.4) where h p l q j : “ T lN p j, j q is given explicitly for t l { u ă j ă N ´ t l { u by h p l q j p b , a q “ ÿ p n , k qP A p l q ρ p l q p n , k q t l { u ´ ź i “´ t l { u b n i j ` i t l { u ´ ź i “´ t l { u ` a k i j ` i . (2.5) Here A p m q is the set A p l q : “ ! p n , k q P N Z ˆ N Z : t l { u ´ ÿ i “´ t l { u p n i ` k i q “ l, @ i ě , n i “ ñ n i ` “ k i ` “ , @ i ă , n i ` “ ñ n i “ k i “ ) . (2.6) The quantity N “ N Y t u and ρ p l q p n , k q P N is given by ρ p l q p n , k q : “ ˆ n ´ ` n ` k k ˙ˆ n ´ ` n n ˙ t l { u ´ ź i “´ t l { u i ‰´ ˆ n i ` n i ` ` k i ` ´ k i ` ˙ˆ n i ` n i ` ´ n i ` ˙ . (2.7) Remark 2.3.
Formula (2.5) holds for t l { u ă j ă N ´ t l { u , for the other values of j the formulais slightly different. This is because for j ď t l { u or j ě N ´ t l { u the polynomial h p l q j is relatedto a constrained Super Motzkin path, [17], instead for t l { u ă j ă N ´ t l { u it is related to aclassical Super Motzkin path. In any case the polynomial h p l q j is independent from N for all j .
6e remark that both | A l | and ρ p l q p n , k q do not depend on N and j . Moreover, from thecondition ř t l { u ´ i “´ t l { u p n i ` k i q “ l in (2.6) one gets that l even ùñ h p l q j contains only even polynomials in a ,l odd ùñ h p l q j contains only odd polynomials in a . (2.8)To prove the almost sure convergence of the empirical spectral distributions d ν p N q H , d ν p N q L and d ν p N q J to their corresponding mean density of states, we will use two general results. The firstone is the following Theorem proved in [18]: Theorem 2.4. (cf. [18, Theorem 2.2]) Consider a random Jacobi matrix T N (1.1) and assumethat t a n u Nn “ and t b n u N ´ n “ have all finite moments. Then for any non trivial polynomial P p x q : A d ν p N q T , P p x q E a.s. Ñ @ d ν T , P p x q D as N Ñ 8 (2.9) ? N ´A d ν p N q T , P p x q E ´ @ d ν T , P p x q D¯ d Ñ N p , σ P q as N Ñ 8 , (2.10) for some constant σ P ě . Here a.s. Ñ is the almost sure convergence and d Ñ is the convergence indistribution. We observe that Theorem 2.4 is not stated in the present form in [18] but this formulationis more convenient for our analysis. The second result is the following classical Lemma whoseproof can be found in [2, 8]:
Lemma 2.5. (cf. [8, Lemma 2.2]) Consider a sequence of random probability measures t d µ n u n “ and d µ a probability measure determined by its moments according to Lemma 2.1. Assume thatany moment of d µ n converges almost surely to the one of d µ . Then as n Ñ 8 the sequence ofmeasures t d µ n u n “ converges weakly, almost surely, to d µ , namely for all bounded and continuousfunctions f : x d µ n , f y Ñ x d µ, f y a.s as n Ñ 8 . (2.11) The convergences still holds for a continuous function f of polynomials growth. Finally, before moving to the actual proof of our main theorem, we summarize the mainresults of [9, 23, 24] in the following theorem.
Theorem 2.6. As N Ñ 8 , βN Ñ α P p , , NM Ñ γ P p , q , a ` α ą , b ` α ą and a R N , the mean spectral measure and the mean density of state of the Gaussian, Laguerre andJacobi β -ensembles weakly converge to the non random measures with density µ α p x q , µ α,γ p x q and µ α,a,b p x q defined in (1.7) , (1.8) and (1.9) respectively. Moreover (2.2) holds for their momentssequences. We are now in position to prove our main result. First of all we remark that the density B α p αµ α p x qq , B α p αµ α,γ p x qq and B α p αµ α,a,b p x qq define a probability measure since the densities µ α p x q , µ α,γ p x q and µ α,a,b p x q define a probability measure. Then, since we want to apply themoment method, we have to compute the moments of the α -ensembles explicitly. To concludethe proof we will need also an explicit expression of the moments of the mean density of statesof the β -ensembles. The following lemma lays the ground to conclude both computations.7 emma 3.1. Fix α P R ` zt u , γ P p , q , a, b ą ´ and N { ą l P N . Consider the α and β -ensembles in Table 1-2, there exist polynomials w l p x q , g l p x q , and rational and continuousfunctions r l p x q such that, for N large enough and βN “ α, NM “ γ , the following holds, for t l { u ă j ă N ´ t l { u : E ” h p l q j ı H β “ w l ` α ` ´ jN ˘˘ ` O ` N ´ ˘ l even l odd , (3.1) E ” h p l q j ı H α “ w l p α q l even l odd , (3.2) E ” h p l q j ı L β “ g l ˆ α ˆ ´ jN ˙˙ ` O ` N ´ ˘ , (3.3) E ” h p l q j ı L α “ g l p α q , (3.4) E ” h p l q j ı J β “ r l ˆ α ˆ ´ jN ˙˙ ` O ` N ´ ˘ , (3.5) E ” h p l q j ı J α “ r l p α q . (3.6) Proof of Lemma 3.1.
We will just prove (3.1)-(3.2) since the proof of the other cases is similar.Indeed the only difference in the proofs is that for the Gaussian and Laguerre α and β -ensembleswe use the fact that the expected value of any even monomial with respect to a χ ξ -distributionis a monomial in ξ . While for the Jacobi α and β -ensembles we use the fact that the expectedvalues of any monomial with respect to a Beta p a, b q -distribution is a rational functions of theparameters.First of all, since a “ p a , . . . , a N q are normal distributed for both ensembles and thanks to(2.8) we get that E ” h p l q j ı H α “ E ” h p l q j ı H β “ , l odd . For the Gaussian α ensemble we have that, for t l { u ă j ă N ´ t l { u E ” h p l q j ı H α “ E »– ÿ p n , k qP A p l q ρ p l q p n , k q t l { u ´ ź i “´ t l { u b n i j ` i t l { u ´ ź i “´ t l { u ` a k i j ` i fifl H α is independent from j since b j ` i „ χ α , a i „ N p , q , i “ ´ t l { u , . . . , t l { u , and the coefficients ρ p l q p n , k q and the set A p l q are independent from j and N by Theorem 2.2. Moreover, as alreadypointed out, the expected values of any even monomial with respect to a χ ξ -distribution is amonomial in ξ . Thus we have that for fixed l P N , there exists a polynomial w l p α q such that(3.2) holds.We can apply a similar reasoning for the Gaussian β ensemble, indeed we notice that ifwe approximate the distribution of b j ` i „ χ α p ´ j ` iN q , i “ ´ t l { u , . . . , t l { u with the one of b j „ χ α p ´ jN q we get an error of order N ´ when we evaluate the expected value. So we can8ompute E ” h p l q j ı H β “ E »– ÿ p n , k qP A p l q ρ p l q p n , k q t l { u ´ ź i “´ t l { u b n i j ` i t l { u ´ ź i “´ t l { u ` a k i j ` i fifl H β “ w l ˆ α ˆ ´ jN ˙˙ ` O ` N ´ ˘ , (3.7)where the only difference from the previous case is that the parameter of the χ -distributionis α ` ´ jN ˘ instead of α .Using the above lemma we can conclude the computation of the moments for the α and β -ensembles: Corollary 3.2.
Fix l P N , α P R ` zt u , a, b ą ´ and γ P p , q then in the large N limit, with N β Ñ α and NM Ñ γ , the following holds: u p l q α : “ lim N Ñ8 E „ N Tr p H lβ q H β “ w l p αx q d x l even l odd , (3.8) v p l q α : “ lim N Ñ8 E „ N Tr p H lα q H α “ w l p α q l even l odd , (3.9) u p l q α,γ : “ lim N Ñ8 E „ N Tr p L lβ q L β “ ż g l p αx q d x , (3.10) v p l q α,γ : “ lim N Ñ8 E „ N Tr p L lα q L α “ g l p α q , (3.11) u p l q α,a,b : “ lim N Ñ8 E „ N Tr p J lβ q J β “ ż r l p αx q d x , (3.12) v p l q α,a,b : “ lim N Ñ8 E „ N Tr p J lα q J α “ r l p α q . (3.13) Proof.
We will just prove (3.8)-(3.9) since the proof of the other cases is analogous.From Lemma 3.1 and Theorem 2.2 one gets that: v p l q α “ lim N Ñ8 »– N ¨˝ N ´ t l { u ´ ÿ j “ t l { u ` w l p α q ` O p q ˛‚fifl “ w l p α q . (3.14)Indeed neglecting the terms h p l q j j “ , . . . , t l { u , N ´ t l { u , . . . , N in the average of Tr ` H lα ˘ weget an error of order O p q since l is fixed, so in the summations we are neglecting a finite numberof terms of order O p q , see Remark 2.3.For the same reason one gets that: u p l q α “ lim N Ñ8 »– N N ´ t l { u ´ ÿ j “ t l { u ` w l ˆ α ˆ ´ jN ˙˙ ` O p N ´ q fifl . (3.15)Thus taking the limit for N going to infinity one gets the integral in (3.8).9 emark 3.3. We stress that u p l q α , u p l q α,γ and u p l q α,a,b are respectively the l th moments of the Gaus-sian, Laguerre and Jacobi β -ensembles in the high temperature regime. Analogously, the quanti-ties v p l q α , v p l q α,γ and v p l q α,a,b are the l th moments of the Gaussian, Laguerre and Jacobi α -ensemblesrespectively. We can now finish the proof of Thereon 1.1
Proof of Theorem 1.1.
From Corollary 3.2 one concludes that for all fixed l P N : v p l q α “ B α p αu p l q α q , (3.16) v p l q α,γ “ B α p αu p l q α,γ q , (3.17) v p l q α,a,b “ B α p αu p l q α,a,b q . (3.18)By Theorem 2.6, Corollary 3.2 and Remark 3.3, the quantities u p l q α , u p l q α,γ and u p l q α,a,b are themoments of the measures with density µ α , µ α,γ and µ α,a,b defined in (1.4), (1.5) and (1.6).Moreover by formula (2.2) such moments uniquely determine the corresponding measures.It follows from relation (3.16) and Lemma 2.1 that the mean density of states d ν H of theGaussian α -ensemble coincides with B α p αµ α q with µ α as (1.4). In a similar way, by (3.17), themeasure B α p αu α,γ q in (1.5) is the mean density of states d ν L of the Laguerre α -ensembles and B α p αu α,a,b q in (1.6) is the mean density of states d ν H of the Jacobi α -ensembles.Since for the α -ensembles all t a n u Nn “ and t b n u N ´ n “ have all finite moments, one can applyTheorem 2.4 getting that the moments of the empirical spectral distributions of the α - ensembles d ν p N q H , d ν p N q L and d ν p N q J converge almost surely to the ones of the corresponding mean density ofstates d ν H , d ν L and d ν J in (1.4), (1.5) and (1.6) respectively. Furthermore applying Lemma 2.5one obtains that the spectral distributions of the α -ensembles d ν p N q H , d ν p N q L and d ν p N q J convergealmost surely to d ν H , d ν L and d ν J in (1.4)-(1.5) and (1.6) respectively.Finally from (2.10) one gets that formula (1.13)–(1.15) hold, namely that the global fluctua-tions are Gaussian. In this section we study the behavior of α -ensembles when the parameter α goes to infinity.For this purpose we consider the rescaled version of α -ensembles, i.e. the matrices defined as ? α H α , γα L α,γ and J α . The corresponding mean density of states is rescaled to d ν H p? αx q , d ν L ´ αxγ ¯ and d ν J p x q (see (1.4)–(1.6)).Now we have to compute the limits of these measures when α Ñ 8 . We will compute theselimits using the matrix representations of the normalized α -ensemble and exploit the followingweak limits: lim α Ñ8 N p , q? α d Ñ α Ñ8 χ α ? α d Ñ α Ñ8 Beta p α, α q d Ñ . (3.19)The above relations imply that the mean density of states of the three normalized α -ensemblesweakly converges to the mean density of states of the following matrices:10 “ ¨˚˚˚˚˚˚˝ . . . . . . . . .. . . . . .
11 0 ˛‹‹‹‹‹‹‚ , L “ ¨˚˚˚˚˚˚˝ ? γ ? γ ` γ ? γ . . . . . . . . .. . . . . . ? γ ? γ ` γ ˛‹‹‹‹‹‹‚ ,J “ ¨˚˚˚˚˚˚˚˝ ? ? ? . . . . . .. . . . . . ˛‹‹‹‹‹‹‹‚ . (3.20)The eigenvalues distributions of the above matrices in the large N limit are given by lim α Ñ8 d ν H p? αx q “ p´ , q π ? ´ x d x , (3.21) lim α Ñ8 d ν L ˆ αxγ ˙ “ pp ´? γ q , p `? γ q q π a γ ´ p x ´ ´ γ q d x , (3.22)(3.23) lim α Ñ8 d ν J p x q “ p , q π a ´ p x ´ q d x , (3.24)where p a,b q is the indicator function of the interval p a, b q .We observe that for all the three α -ensembles in the large α limit, the corresponding meandensity of states is an arcsine distribution. It would be interesting to study the behavior of thefluctuations of the max/min eigenvalue of the α -ensembles in the limit of large α . In this section we will apply Theorem 1.1 to find the mean density of states of the classical Todachain [22] with periodic boundary conditions. As we already mentioned this is an alternativeproof of the result in [21].
The classical Toda chain is the dynamical system described by the following Hamiltonian: H T p p , q q : “ N ÿ j “ p j ` N ÿ j “ V T p q j ` ´ q j q , V T p x q “ e ´ x ` x ´ , (4.1)with periodic boundary conditions q j ` N “ q j @ j P Z . Its equations of motion take theform q j “ B H T B p j “ p j , p j “ ´ B H T B q j “ V T p q j ` ´ q j q ´ V T p q j ´ q j ´ q , j “ , . . . , N . (4.2)11t is well known that the Toda chain is an integrable system [13, 22], one way to prove it is toput the Toda equations in Lax pair form. This was introduced by Flaschka [11] and Manakov [16]through the following non canonical change of coordinates: a j : “ ´ p j , b j : “ e p q j ´ q j ` q ” e ´ r j , ď j ď N , (4.3)where r j “ q j ` ´ q j is the relative distance. The periodic boundary conditions imply N ÿ j “ r j “ . (4.4)Then, defining the Lax operator L as the periodic Jacobi matrix [25] L p b , a q : “ ¨˚˚˚˚˚˚˚˝ a b . . . b N b a b . . . ... b a . . . ... . . . . . . . . . b N ´ b N . . . b N ´ a N ˛‹‹‹‹‹‹‹‚ , (4.5)and the anti-symmetric matrix BB p b q : “ ¨˚˚˚˚˚˚˚˝ b . . . ´ b N ´ b b . . . ... ´ b . . . ... . . . . . . . . . b N ´ b N . . . ´ b N ´ ˛‹‹‹‹‹‹‹‚ , (4.6)a straightforward calculation shows that the equations of motions (4.2) are equivalent to dLdt “ r B ; L s , (4.7)so the eigenvalues of L are a set of integrals of motion. We consider the evolution of the Toda chain on the subspace: M : “ p p , r q P R N ˆ R N : N ÿ j “ r j “ N ÿ j “ p j “ + , (4.8)which is invariant for the dynamics. Indeed the condition ř Nj “ r j “ follows from the periodicboundary conditions and the condition ř Nj “ p j “ follows from the fact that the system istranslational invariant and therefore the total momentum is conserved. We endow the phasespace M (4.8) with the Gibbs measure for the Toda lattice at temperature β ´ as d ν T oda : “ Z T oda p β q e ´ βH T p p , r q δ ř Nj “ p j δ ř Nj “ r j d p d r , (4.9)12ere Z T oda p β q is the partition function which normalize the measure.We notice that this ensemble makes L (4.5) into a random matrix, thus it makes sense tostudy its mean density of states. However the matrix entries of L are not independent randomvariables because of (4.8). For this reason we also introduce the approximate measure d r ν T oda on R N ˆ R N as d r ν T oda : “ r Z T oda p β q e ´ βH T p p , r q´ θ ř j r j d p d r , (4.10)where r Z T oda p β q is the partition function which normalizes the measure and θ ą is chosen insuch a way that: x r j , d r ν T oda y “ . (4.11)The value of θ ą is unique for all β ą since x r j , d r ν T oda y “ log p β q ´ Γ p β ` θ q Γ p β ` θ q , (4.12)which has just one positive solution.From now on we will write L and r L as the random matrices whose entries are distributedaccording to the probability measure d ν T oda and d r ν T oda respectively. In particular applying thechange of coordinates (4.3) one gets that r L „ ? β ¨˚˚˚˚˚˚˚˝ a b . . . b N b a b . . . ... b a . . . ... . . . . . . . . . b N ´ b N . . . b N ´ a N ˛‹‹‹‹‹‹‹‚ , b j „ χ p β ` θ q , a j „ N p , q j “ , . . . , N. (4.13)To obtain the mean density of states of the Toda lattice with periodic boundary conditionswe need the following lemma, whose proof can be found in [12]: Lemma 4.1. (cf. [12, Lemma 4.1]) Fix r β ą and let f : R N ˆ R N Ñ R depend on just K variables and finite second order moment with respect to d r ν T oda , uniformly for all β ą r β . Thenthere exist positive constants C, N and β such that for all N ą N , β ą max t β , ˜ β u one has |x f, d ν T oda y ´ x f, d r ν T oda y| ď
C KN a x f , d r ν T oda y ´ x f, d r ν T oda y . (4.14)Applying this Lemma we can conclude that the matrices L and r L have the same momentsequence in the large N limit. Furthermore, r L is a rank one perturbation of the matrix ? β H θ ` β in table 1. So we can use the following theorem, whose proof can be found in [4], to show thatthe mean density of states of the matrices r L and ? β H θ ` β in the large N limit are the same. Theorem 4.2. (cf. [4, Theorem A.43]) Let
A, B be two N ˆ N Hermitian matrices and F A , F B their empirical spectral density defined as: F A p x q : “ N t j ď N : λ j ď x u , (4.15) where λ j are the eigenvalues of A . Then | F A ´ F B || ď N Rank p A ´ B q , (4.16) where || f || “ sup x | f p x q| . This implies also that the moment sequence of r L and ? β H θ ` β are the same in the large N limit, which means that also the moment sequence of L, ? β H θ ` β in the large N limit are equal.So applying Lemma 2.1 and Theorem 1.1 one gets that : Lemma 4.3.
Consider the classical Toda chain (4.1) and endow the phase space M (4.8) withthe Gibbs measure d ν T oda in (4.9) , then there exists a constant β ą such that, for all β ą β the mean density of states of the Lax matrix L (4.5) in the limit N Ñ 8 is explicitly given by: d ξ L p x q “ a β B α p αµ α p a βx qq | α “ β ` θ d x , (4.17) where µ α p x q is given in (1.7) . To conclude, we also remark that if we let the inverse temperature β approach infinity, inview of (3.21), we obtain that the mean density of states of the classical Toda chain in thisregime is exactly the arcsine law (1.4). From the physical point of view the system is at rest. Acknowledgments.
This project has received funding from the European Union’s H2020research and innovation program under the Marie Skłowdoska–Curie grant No. 778010
IPaDE-GAN and from LIA, LYSM, AMU, CNRS, ECM, INdAM.14 = 1 = 10 = 50 = 100
Figure 1: Gaussian α ensemble empirical spectral density for different values of the parameters, N “ , trials: . 15 = 1, = 0.8 = 10, = 0.8 = 50, = 0.8 = 100, = 0.8 Figure 2: Laguerre α ensemble empirical spectral density for different values of the parameters, N “ , trials: . 16 .3 0.4 0.5 0.6 0.7 0.8 0.90.00.51.01.52.02.53.0 = 1, a = 25.8, b = 10 = 10, a = 25.8, b = 10 = 50, a = 25.8, b = 10 = 100, a = 25.8, b = 10 Figure 3: Jacobi α ensemble empirical spectral density for different values of the parameters, N “ , trials: . 17 Associate Orthogonal polynomials
The associate Hermite polynomials H p α q n p x q were introduced in [3]. They are orthonormal poly-nomials with respect to the measure µ α defined in (1.7), namely ż `8´8 H p α q n p x q H p α q n p x q µ α p x q dx “ δ nm , and satisfy the following three terms recurrence relation: xH p α q n p x q “ H p α q n ` p x q ` p n ` α q H p α q n ´ p x q , H ´ p x, α q “ , H p x q “ , (A.1)for α “ one gets the standard Hermite polynomials.The associate Laguerre polynomials of type 2, L α,γn p x q , were introduced [14]. They satisfythe orthogonality relation ż L α,γn p x q L α,γm p x q µ α,γ p x q dx “ δ nm , where µ α,γ p x q “ p α ` q Γ ´ ` αγ ` α ¯ x αγ e ´ x (cid:12)(cid:12)(cid:12) ψ ´ α, ´ αγ ; xe ´ iπ ¯ (cid:12)(cid:12)(cid:12) , is defined in (1.8). They also satisfy the following three terms recurrence relation: L p x q “ , L p x q “ α ` αγ ` ´ xα ` , (A.2) ´ xL α,γn p x q “ p n ` ` α q L α,γn ` ´ ˆ n ` αγ ` α ` ˙ L α,γn p x q ` ˆ n ` α ` αγ ˙ L α,γn ´ p x q . (A.3)In the definition of µ α,γ the Tricomi confluent hypergeometric function ψ p a, b ; z q [1] is definedto be the standard solution of the Kummer’s equation z d ψ d z ` p b ´ z q d ψ d z ´ aψ “ , (A.4)uniquely determined by the normalization ψ p a, b ; z q „ z ´ a as z Ñ 8 and | arg p z q| ď π , here arg p z q is the argument of the complex number z . Moreover if b R N then there exists an alternativeformula for the Tricomi confluent hypergeometric: ψ p a, b ; z q “ Γ p ´ b q Γ p a ´ b ` q F p a, b ; z q ` Γ p b ´ q Γ p a q F p a ´ b ` , ´ b ; z q , (A.5)where F p a, b ; z q “ ÿ n “ p a q n p b q n n ! z n , p a q n “ a p a ` q ¨ ¨ ¨ p a ` n ´ q . (A.6)It was shown in [24] that µ α,a,b (1.9) is the orthogonality measure of the associate Jacobipolynomials of type 3, J α,a,bn p x q : ż J α,a,bn p x q J α,a,bm p x q µ α,a,b p x q dx “ δ nm . J α,a,bn p x q satisfy the following recurrence relation: xJ α,a,bn p x q “ a ξ n µ n ` J α,a,bn ` p x q ` p ξ n ` η n q J α,a,bn p x q ` a ξ n ´ µ n J α,a,bn ´ p x q , (A.7)where $’&’% ξ p α q “ α ` a ` α ` a ` b ` ξ n p α q “ n ` α ` a ` n ` α ` a ` b ` n ` α ` a ` b ` n ` α ` a ` b ` , n ą η n p α q “ n ` α n ` α ` a ` b ` n ` α ` b n ` α ` a ` b , n ą , α ě , a, b ą ´ , (A.8)19 eferences [1] M. Abramowitz and I. A. Stegun , eds.,
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