aa r X i v : . [ m a t h . P R ] J a n Three-Parametric Marcenko–Pastur Density
Taiki Endo ∗ , Makoto Katori † Abstract
The complex Wishart ensemble is the statistical ensemble of M × N complex random matriceswith M ≥ N such that the real and imaginary parts of each element are given by independentstandard normal variables. The Marcenko–Pastur (MP) density ρ ( x ; r ) , x ≥ N → ∞ , M → ∞ with a fixed rectangularity r = N/M ∈ (0 , ρ ( x ; r, t ) with time t ≥
0, whose initial distribution is δ ( x ). Recently, Blaizot, Nowak, and Warcho l studied thetime-dependent complex Wishart ensemble with an external source and introduced the three-parametric MP density ρ ( x ; r, t, a ) by analyzing the hydrodynamic limit of the process startingfrom δ ( x − a ) , a >
0. In the present paper, we give useful expressions for ρ ( x ; r, t, a ) and performa systematic study of dynamic critical phenomena observed at the critical time t c ( a ) = a when r = 1. The universal behavior in the long-term limit t → ∞ is also reported. It is expectedthat the present system having the three-parametric MP density provides a mean-field modelfor QCD showing spontaneous chiral symmetry breaking. Keywords
Marcenko–Pastur law · Wishart random-matrix ensemble · Wishart process · Random-matrix ensemble with an external source · Hydrodynamic limit · Dynamic criticalphenomena · Spontaneous chiral symmetry breaking
Assume that
M, N ∈ N := { , , . . . } , M ≥ N . Consider M × N complex random matrices K = ( K jk ) such that the real and the imaginary parts of elements are i.i.d. and normally distributedwith mean µ = 0 and variance σ = 1 /
2. The normal distribution with mean µ and variance σ isdenoted by N ( µ, σ ) and when a random variable X obeys N ( µ, σ ), we write it as X ∼ N ( µ, σ ).Then the present setting is described as ℜ K jk ∼ N (0 , / , ℑ K jk ∼ N (0 , / , j = 1 , . . . , M, k = 1 , . . . , N. ∗ Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: [email protected] † Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan; e-mail: [email protected]
1e consider a statistical ensemble of N × N Hermitian random matrices L defined by L = K † K, (1.1)where K † denotes the Hermitian conjugate of K . This ensemble of random matrices is called the complex Wishart random-matrix ensemble or the chiral Gaussian unitary ensemble (chGUE) (see,for instance, [8]). We denote the eigenvalues of L as X Nj , j = 1 , . . . , N , which are nonnegative,since L is nonnegative definite by definition; X Nj ∈ R ≥ := { x ∈ R : x ≥ } . The positive squareroots of them, q X Nj , j = 1 , . . . , N are called singular values of random rectangular matrices K .In other words, the eigenvalue distribution of the Hermitian random matrices L can be regardedas the distribution of squares of singular values of the rectangular complex random matrices K inthe complex Wishart random-matrix ensemble.Let C c ( R ) be the set of all continuous real-valued function with compact support on R . Weconsider the empirical measure defined byΞ N ( dx ) = 1 N N X j =1 δ X Nj /M ( dx ) , x ∈ R ≥ , (1.2)where δ y ( dx ) denotes a Dirac measure concentrated on y such that R R f ( x ) δ y ( dx ) = f ( y ) for all f ∈ C c ( R ). Then we take the double limit N → ∞ , M → ∞ for each fixed value of the rectangularity r = lim N →∞ ,M →∞ NM ∈ (0 , . (1.3)We can prove that in this scaling limit (1.3), the empirical measure (1.2) converges weakly to adeterministic measure ρ ( x ) dx, x ∈ R ≥ in the sense that R R ≥ f ( x )Ξ N ( dx ) → R R ≥ f ( x ) ρ ( x ) dx as N → ∞ for any f ∈ C c ( R ). Moreover, the probability density ρ in the limit measure has a finitesupport in R and it is explicitly given as a function of the parameter r ∈ (0 ,
1] as [18] ρ ( x ; r ) = p ( x − x L ( r ))( x R ( r ) − x )2 πrx ( x L ( r ) ,x R ( r )) ( x ) (1.4)with x L ( r ) := (1 − √ r ) , x R ( r ) := (1 + √ r ) . (1.5)Here Λ ( x ), Λ ⊂ R is an indicator function such that Λ ( x ) = 1 if x ∈ Λ, and Λ ( x ) = 0 otherwise.This convergence theorem is known as the Marcenko–Pastur law for the Wishart random-matrixensemble [18, 8, 1] and we call (1.4) the
Marcenko–Pastur (MP) density in this paper.
A dynamical extension of the eigenvalue distribution of the Wishart random-matrix ensemble isrealized by the solution { X Nj ( t ) ∈ R ≥ : t ≥ , j = 1 , , . . . , N } of the following system of stochasticdifferential equations (SDEs), dX Nj ( t ) = 2 q X Nj ( t ) dB j ( t ) + 2( ν + 1) dt + 4 X Nj ( t ) X ≤ k ≤ N,k = j X Nj ( t ) − X Nk ( t ) dt, j = 1 , , . . . , N, t ≥ , (1.6)2here ν = M − N and B j ( t ) , t ≥ x Nj ∈ R ≥ , j = 1 , . . . , N . We assume that 0 ≤ x N ≤ x N ≤ · · · ≤ x NN < ∞ . Thisone-parameter family ( ν >
0) of N -particle stochastic processes was called (the eigenvalue processof) the Wishart process by Bru [5]. It is also called the
Laguerre process or the noncolliding squaredBessel process [15, 13, 14].We set ν = (1 − r ) M = (1 − r ) N/r , r ∈ (0 , Nt ( dx ) := 1 N N X j =1 δ X Nj ( t ) /M ( dx ) , x ∈ R ≥ , t ≥ . (1.7)If the initial empirical measure satisfies some moment conditions and converges weakly to a measure,Ξ N ( dx ) = (1 /N ) P Nj =1 δ x Nj /M ( dx ) → ξ ( dx ) in the limit N → ∞ , M → ∞ with r = N/M fixed in(0 , Nt ( dx ) converges weakly to a time-dependent deterministic measure, whichwe denote here as ρ ξ ( x ; r, t ) , t ≥ Green’s function (the resolvent) G ξ ( z ; r, t )by the Stieltjes transform of ρ ξ , G ξ ( z ; r, t ) := Z R ρ ξ ( x ; r, t ) z − x dx, z ∈ C \ R . Then we can prove that this solves the following nonlinear partial differential equation (PDE) [6, 2], ∂G ξ ∂t = − ∂G ξ ∂z + r (cid:26) ∂G ξ ∂z − zG ξ ∂G ξ ∂z − G ξ (cid:27) , t ∈ [0 , ∞ ) , (1.8)under the initial condition, G ξ ( z ; r,
0) = R R ξ ( dx ) / ( z − x ), z ∈ C \ R . Once the Green’s function G ξ ( z ; r, , z ∈ C \ R is determined, we can obtained the density function following the Sokhotski-Plemelj theorem, ρ ξ ( x ; r, t ) = −ℑ (cid:20) lim ε → π G ξ ( x + iε ; r, t ) (cid:21) , i := √− . (1.9)We regard (1.8) as the analogy of the complex Burgers equations in the inviscid limit ( i.e. , the(complex) one-dimensional Euler equation), and we call the limit process the hydrodynamic limit of the Wishart process [2, 3].The simplest case of the dynamical extension of the MP density (1.4) with (1.5) is obtained bysetting the initial distribution, ξ ( dx ) = δ ( dx ) := δ ( x ) dx, (1.10)that is, all particles are concentrated on the origin; x Nj = 0 , j ∈ N . By the method of com-plex characteristics, Blaizot, Nowak, and Warcho l [2] showed that the Green’s function for (1.10), G δ ( z ) = G δ ( z ; r, t ), is given by the solution of the equation z = 1 G δ ( z ) + t − rtG δ ( z ) , z ∈ C \ R , r ∈ (0 , , t ≥ . (1.11)They solved (1.11) and using the Sokhotski-Plemelj theorem (1.9) derived the time-dependentextension of (1.4), ρ ( x ; r, t ) := ρ δ ( x ; r, t ) = p ( x − x L ( r, t ))( x R ( r, t ) − x )2 πrtx ( x L ( r,t ) ,x R ( r,t )) ( x ) (1.12)3ith x L ( r, t ) := (1 − √ r ) t, x R ( r, t ) := (1 + √ r ) t, t ∈ (0 , ∞ ) . (1.13)Since ρ ( x ; r,
1) is equal to the original MP density (1.4), the above provides a dynamical derivationof the Marcenko–Pastur law. The dependence on t of ρ ( x ; r, t ) given by (1.12) with (1.13) is verysimple, but we regard this as the two-parametric MP density in the present paper.
Blaizot, Nowak, and Warcho l [3] have studied the hydrodynamic limit of the Wishart eigenvalue-process starting from one-parameter family ( a ≥
0) of the initial distribution, ξ ( dx ) = δ a ( dx ) := δ ( x − a ) dx. They showed that the Green’s function, G ( z ) = G δ a ( z ; r, t ), a ≥
0, is obtained by the solution ofthe equation, z = 1 G δ a ( z ) + t − rtG δ a ( z ) + a (1 − rtG δ a ( z )) , z ∈ C \ R , r ∈ (0 , , t ≥ , a ≥ . (1.14)They claimed in [3] that a proper solution of this equation yields ρ δ a ( x ; r, t ) via the Sokhotski-Plemelj theorem and showed an illustration (Fig.1 in [3]) of the time dependence of this densityfunction for a special case with r = 1 and a = 1. The explicit formula of ρ δ a ( x ; r, t ) was, however,not given there. See also Section 8 in [17] and Section 3 in [10] for implicit expressions of G δ a .We write the solution discussed in [3] as ρ ( x ; r, t, a ) and call it the three-parametric Marcenko-Pastur (MP) density . The purpose of the present paper is to report useful expressions and detailedanalysis of this density function ρ ( x ; r, t, a ) on R ≥ with three parameters r ∈ (0 , , t ≥ a ≥ Theorem 1.1
Let S ( x ; r, t, a ) := 4 ax − { a + 4 a (3 r + 2) t − t } x + 2[2 a − a (5 r − t + a { r (6 r −
1) + 1 } t − ( r + 1) t ] x + ( r − t { a − a (4 r − t + t } . (1.15) For r ∈ (0 , , t > , a ≥ , consider the case such that the cubic equation with respect to x , S ( x ; r, t, a ) = 0 , (1.16) has three real solutions, x ≤ x ≤ x , where x j = x j ( r, t, a ) , j = 1 , , . Define x L ( r, t, a ) := x ( r, t, a ) , x R ( r, t, a ) := x ( r, t, a ) . (1.17) Put g = g ( x ; r, t, a ):= − x + 3 { (2 r + 1) t + 6 a } x − h ( r − { (2 r + 1) t − a } t − √− S i x + 2( r − t , (1.18)4 ith S = S ( x ; r, t, a ) given by (1.15), and define ϕ = ϕ ( x ; r, t, a ):= − { x − ( r − t } − / x + { a − (2 r + 1) t } x + t ( r − g / − g / × / . (1.19) Then the three-parametric MP density is given by ρ ( x ; r, t, a ) = p ( x − f L ( x ; r, t, a ))( f R ( x ; r, t, a ) − x )2 πrxt ( x L ( r,t,a ) ,x R ( r,t,a )) ( x ) (1.20) with f L ( x ; r, t, a ) := p d − + p d + − p d ! , f R ( x ; r, t, a ) := p d − + p d + p d ! , (1.21) where d − = t − a − √ aϕ, d + = t − a + 2 √ aϕ,d = ϕ + x + t − a p d − d + . (1.22) Remark 1
The formula (1.20) for the present three-parametric MP density seems to be similar tothe original MP density (1.4) and the two-parametric MP density (1.12). We should note, however,that f L and f R appearing in (1.20) are not equal to the endpoints x L and x R of the support ofdensity and they depend on x as shown by (1.21) with (1.18), (1.19) and (1.22). We can see that ϕ ( x ; r, t,
0) := lim a → ϕ ( x ; r, t, a ) = − x + ( r − t, and hence d ± → t, d → rt as a →
0. Then as a → f L → x L ( r ; t ), f R → x R ( r ; t ) with (1.13); thatis, the dependence of f L and f R on x vanishes only in this limit. Theorem 1.1 states that for general a >
0, the endpoints x L and x R of the support for the three-parametric MP density are given bythe suitably chosen solutions (1.17) of the cubic equation (1.16) with (1.15) as proved in Section2.2 below. That is, the formula (1.20) is universal, but the choice of solutions (1.17) depends onthe parameters r, t, a . The equation (1.14) seems to be a simple perturbation of (1.11), but thesolution turns out to have rich structures, by which we can describe dynamic critical phenomenaat time t = t c := a for a >
0, when r = 1, as shown below.From the view point of the original random matrix theory, Theorem 1.1 gives the limit theoremfor the eigenvalue distribution of random matrix L given by (1.1) in the scaling limit (1.3), in which M × N rectangular complex random matrices K = ( K jk ) are distributed as ℜ K kk ∼ N ( √ M a, t/ , k = 1 , . . . , N, ℜ K jk ∼ N (0 , t/ , j = 1 , . . . , M, k = 1 , . . . , N, j = k, ℑ K jk ∼ N (0 , t/ , j = 1 , . . . , M, k = 1 , . . . , N, (1.23) t > , r ∈ (0 , E [ L jk ] = M ( a + t ) δ jk , j, k = 1 , . . . , N. t, a ) = (1 ,
0) and (1 ,
1) are superposed in order to compare each other. They show the distributionsof the eigenvalues of L = K † K given by K of size 1000 × t, a ) = (1 ,
0) and (1 , ρ ( x ; r = 0 .
3) and the three-parametric MP density ρ ( x ; r = 0 . , t = 1 , a = 1) are shownby a thin curve and a thick curve, respectively. Due to an external source at x = a = 1, theeigenvalue distribution with ( t, a ) = (1 , ρ ( x ; r = 0 . , t = 1 , a = 1), is shifted to the positive direction and becomes broader compare withthe original MP density ρ ( x ; r = 0 . a >
0, such an ensemble of random matrices will be called the
Wishart ensemble with anexternal source or the non-centered Wishart ensemble , since even at t = 0, the diagonal elementsof L have positive means, E [ L jj ] = M a > , j = 1 , . . . , N , [4, 16, 7, 11, 9].In Figure 1, we compare two histograms for the empirical measures (1.7) of the eigenvalues ofmatrices L = K † K given by K of size 1000 ×
300 (with the rectangularity r = 300 / . t, a ). When( t, a ) = (1 , x ≃ .
4. When ( t, a ) = (1 , x = a = 1, the distribution is shifted to the positive direction havinga maximum at x ≃ ρ ( x ; r = 0 .
3) and the latter is by the three-parametric MP density ρ ( x ; r = 0 . , t = 1 , a = 1) givenby (1.20).For each values of rectangularity r ∈ (0 ,
1] and strength of an external source a ≥
0, we canshow time evolution of the support ( x L ( r, t, a ) , x R ( r, t, a )) of ρ ( x ; r, t, a ) on the ( x, t )-plane, ( R ≥ ) .Figure 2 shows the domains D ( r, a ) := { ( x L ( r, t, a ) , x R ( r, t, a )) : t ≥ } ⊂ ( R ≥ ) for ( r, a ) = (0 . ,
0) and ( r, a ) = (0 . , r, a ) = (1 ,
0) and ( r, a ) = (1 , a > r = 1.6 xt xt Figure 2: For r = 0 .
3, time evolution of the support ( x L , x R ) is shown on the ( x, t )-plane forthe two-parametric MP density ρ ( x ; r = 0 . , t ) := ρ ( x ; r = 0 . , t, a = 0) in the left, and for thethree-parametric MP density with a = 1, ρ ( x ; r = 0 . , t, a = 1) in the right. The supports areextended in time, but the left edges of supports are kept to be positive, x L >
0, for all t > xt xt Figure 3: For r = 1, time evolution of the support ( x L , x R ) is shown on the ( x, t )-plane for the two-parametric MP density ρ ( x ; r = 1 , t ) := ρ ( x ; r = 1 , t, a = 0) in the left, and for the three-parametricMP density with a = 1, ρ ( x ; r = 1 , t, a = 1) in the right. In the two-parametric MP density, thesupport starts from the singleton { } at t = 0 and the left edge of support x L is identically zero; x L ≡ t ≥
0. On the other hand, in the three-parametric MP density with a = 1, the supportstarts from the singleton { } at t = 0, and x L > t < t c = 1. As t ր t c = 1, however, x L ց x L ≡ t ≥ t c = 1. We regard t c = 1 as a critical time.7 roposition 1.2 Assume that a > . (i) If and only if r = 1 , D ( r, a ) touches the origin x = 0 . Otherwise, the left edge of supp ρ ( x ; r, t, a ) is strictly positive; x L ( r, t, a ) > , r ∈ (0 , . (ii) When r = 1 , there is a critical time t c ( a ) = a such that x L (1 , t, a ) > while ≤ t < t c ( a ) , and x L (1 , t, a ) ≡ for t ≥ t c ( a ) . In particular,just before the critical time t c ( a ) , the left edge of supp ρ ( x ; r, t, a ) behaves as x L (1 , t, a ) ≃ a ( t c ( a ) − t ) ν with ν = 3 as t ր t c ( a ) . In the case with r = 1, the dynamic critical phenomena at the critical time t = t c ( a ) areobserved in the vicinity of the origin as follows. Proposition 1.3
When r = 1 , the three-parametric MP density shows the following dynamiccritical phenomena at t = t c ( a ) . (i) For < t < t c ( a ) , ρ ( x ; 1 , t, a ) ≃ C ( t, a )( x − x L (1 , t, a )) β with β = 12 as x ց x L (1 , t, a ) , where C ( t, a ) ≃ a π ( t c ( a ) − t ) − γ with γ = 52 as t ր t c ( a ) . (ii) At t = t c ( a ) , ρ ( x ; 1 , t c ( a ) , a ) ≃ √ π a − / x − γ with γ = 13 as x ց . (iii) For t > t c ( a ) , ρ ( x ; 1 , t, a ) ≃ C ( t, a ) x − γ with γ = 12 as x ց , where C ( t, a ) ≃ πt c ( a ) ( t − t c ( a )) β with β = 12 as t ց t c ( a ) . Remark 2
The critical exponents ν = 3 and γ = 1 / C of ρ in the subcritical time-region(0 < t < t c ( a )) diverges with the critical exponent γ = 5 / t ր t c ( a ). Then both at the criticaltime ( t = t c ( a )) and in the supercritical time-region ( t > t c ( a )) ρ diverges as x ց
0, but the criticalexponents are different. In the supercritical time-region, the amplitude C of the diverging ρ withthe critical exponent γ = 1 / t ց t c ( a ) (with the exponent β = 1 / .01 0.02 0.03 0.04 0.05 x ρ Figure 4: Critical behavior of the three-parametric MP density ρ is shown for r = 1 and a = 1with the critical time t c (1) = 1. The dashed curve denotes the emergence of ρ at x = x L ≃ . β = 1 / t = 0 . t c (1)). The divergence of ρ as x ց t = t c (1) is shown by a solid curve and that at a supercritical time( t = 1 . t c (1)) by a dotted curve. The former with the critical exponent γ = 1 / γ = 1 / ρ as x ց t = t c ( a ) having a smaller value of exponent as γ = 1 / < γ = 1 /
2. See Fig. 4. The proof of Proposition 1.3 given in Subsection 2.4 implies thescaling relation ν = β + γ . The functions S ( x ; r, t, a ), g ( x ; r, t, a ), ϕ ( x ; r, t, a ), f L ( x ; r, t, a ), and f R ( x ; r, t, a ), which appearin Theorem 1.1, are all homogeneous as multivariate functions of x, t, a for each fixed value of r ∈ (0 , ρ ( κx ; r, κt, κa ) = 1 κ ρ ( x ; r, t, a ) , r ∈ (0 , , (1.24)for an arbitrary parameter κ >
0. By this property, the following long-term behavior of the three-parametric MP density is readily concluded.
Proposition 1.4
For a ≥ , lim t →∞ ρ ( y ; r, t, a ) dy (cid:12)(cid:12)(cid:12) y = tx = ρ ( x ; r ) dx, where ρ ( x ; r ) is given by (1.4) with (1.5). The long-term behavior of the present three-parametric MP density is given by a dilatation of theoriginal MP density by factor t . In this sense, the original Marcenko–Pastur law is universal andit describes the large-scale and long-term behavior of the Wishart ensemble and process.9 - ρ chiral Figure 5: Time evolution of the hydrodynamical density ρ chiral ( x ; r, t, a ) of the QCD Dirac operatorin the critical case, which is obtained as (1.25) from the present three-parametric MP density with r = 1. For a = 1, ρ chiral is plotted for t = 0 . ≤ t ≤ t c = 1, the density at the origin ρ chiral (0; r = 1 , t, a = 1) = 0, while itbecomes positive for t > t c . For t > t c , ρ chiral shows a relaxation in the sense of (1.27) to theuniversal density following Wigner’s semicircle law (1.28). Remark 3
So far we have studied the density ρ of eigenvalues of random matrices L given by(1.1) in the hydrodynamic limit. On the other hand, when the present random matrix ensemble,chGUE, is applied as a model to the quantum chromodynamics (QCD) in high energy physics,the density ρ chiral of the positive-signed and negative-signed singular values of random rectangularmatrix K have been discussed [12, 17, 10]. For the transformation from ρ to ρ chiral , see Eq.(3.34)in [10], for instance. The present three-parametric MP density ρ ( x ; r, t, a ) given by Theorem 1.1provides the following hydrodynamical description of the time-depending spectrum for the QCDDirac operator with parameters r ∈ (0 ,
1] and a ≥ ρ chiral ( x ; r, t, a ) = 2 | x | ρ ( x ; r, t, a ) , x ∈ R , t > , (1.25)under the initial state ρ chiral ( x ; r, , a ) = δ ( x + a ) + δ ( x − a ) , x ∈ R . Figure 5 shows the time evolution of (1.25) in the critical case r = 1 with a = 1. By (1.25),Proposition 1.3 (ii) and (iii) give the following for r = 1, ρ chiral ( x ; 1 , t c ( a ) , a ) ≃ √ π a − / | x | /δ with δ = 3 as | x | → ,ρ chiral (0; 1 , t, a ) ≃ πt c ( a ) ( t − t c ( a )) β with β = 12 as t ց t c ( a ) . (1.26)10oreover, Proposition 1.4 implies through (1.25) that, when r = 1,lim t →∞ ρ chiral ( y ; 1 , t, a ) dy (cid:12)(cid:12)(cid:12) y = √ tx = lim t →∞ ρ ( tx ; 1 , t, a ) d ( tx )= ρ ( x ; 1) dx = 2 xρ ( x ; 1) dx = ρ Wigner ( x ) dx, x ∈ R , (1.27)where ρ Wigner ( x ) = 1 π p − x ( − , ( x ) (1.28)is the density function describing Wigner’s semicircle law (see, for instance, [8]). As mentionedin [17], the time evolution of ρ chiral from the two-peak shape with zero density at the origin (0 ≤ t ≤ t c ( a )) to the universal shape ρ Wigner after t ∼ t c ( a ) via a critical shape at t = t c ( a ) can beinterpreted as a transition from an initial state with restored chiral symmetry to a final state with spontaneous chiral symmetry breaking . In [12], we find the argument that the present system withthe density ρ chiral ( x ; r, t, a ) give a mean-field model for QCD and δ = 3 and β = 1 / For the Green’s function G δ a ( z ) = G δ a ( z ; r, t ), a ≥
0, we put R = R ( x ) := lim ε → ℜ G δ a ( x + iε ) ,I = I ( x ) := − lim ε → ℑ G δ a ( x + iε ) , that is, lim ε → G ( x + iε ) = R ( x ) − iI ( x ). For the three-parametric MP density ρ ( x ) := ρ ( x ; r, t, a ),its Hilbert transform is defined by H [ ρ ]( x ) = 1 π − Z R ρ ( y ) x − y dy := 1 π lim ε → (cid:26)Z x − ε −∞ ρ ( y ) x − y dy + Z ∞ x + ε ρ ( y ) x − y dy (cid:27) . The Sokhotski-Plemelj theorem states that ρ ( x ) = I ( x ) π , H [ ρ ( · )]( x ) = R ( x ) π . (2.1)Let A = 1 R + I , B = 1(1 − rtR ) + ( rtI ) . (2.2)11y definition, we obtain the equation, A = ( rt ) rtR − /B . (2.3)For the equation (1.14), we obtain the following. Lemma 2.1
The equation (1.14) for the complex-valued function G δ a ( z ) , a ≥ is equivalent withthe following system of equations for the real-valued functions A and B , x = RA + (1 − rtR ) tB + a (cid:2) (1 − rtR ) − ( rtI ) (cid:3) B , A − rt B ] − a (1 − rtR ) rtB . (2.4) Proof
We put z = x + iε, x, ε ∈ R in (1.14) and take the limit ε →
0. Then the real part and theimaginary part of the obtained equation give (2.4).Before solving the system of equations (2.4) for general a ≥
0, first we solve it for the specialcase a = 0. In this case (2.4) with (2.1) and (2.2) are simplified as x = R R + ( πρ ) + (1 − rtR ) t (1 − rtR ) + ( rt ) ( πρ ) = (cid:26) R + ( πρ ) − rt (1 − rtR ) + ( rt ) ( πρ ) (cid:27) R + t (1 − rtR ) + ( rt ) ( πρ ) , R + ( πρ ) − rt (1 − rtR ) + ( rt ) ( πρ ) , for R := R ( x ; r, t,
0) and ρ := ρ ( x ; r, t, x , x = 1 rt { R + ( πρ ) } , (2.5) x = t (1 − rtR ) + ( rt ) ( πρ ) . (2.6)From (2.5), we have the relation ( πρ ) = 1 rtx − R . (2.7)Combining this with (2.6), we have x = t/ (1 − rtR + rt/x ), which is solved as R = x + ( r − t rtx . (2.8)Put (2.8) into (2.7), we obtain( πρ ) = 1(2 rtx ) {− x + 2( r + 1) tx − ( r − t } = ( x − x L )( x R − x )(2 rtx ) , (2.9)where x L = x L ( r, t ) and x R = x R ( r, t ) are given by (1.13) and the two-parametric MP density(1.12) is obtained as the positive square root of (2.9) for x L ≤ x ≤ x R .12he above calculation suggests that it will be easier to obtain R than I . By (2.1) and the firstequation of (2.2), if 1 A − R ≥ , then ρ ( x ) = 1 π r A − R . (2.10)Hence if we can express A , not using I , but using only R and parameters r, t, a , then the obtained R determines the density function ρ . Actually we will show that this strategy is successful in thefollowing.By eliminating A in (2.4), we obtain a quadratic equation for B as2 a (1 − rtR ) B + ( t − a ) B − x = 0 . (2.11)We choose the following solution of (2.11), B = a − t + √ D a (1 − rtR ) , (2.12)with D = D ( x ; r, t, a ) = 8 a (1 − rtR ) x + ( t − a ) , (2.13)by the following reason. If we put a = 0, (2.11) gives B | a =0 = x/t . On the other hand, (2.13) gives √ D = − ( a − t ) + 4 a (1 − rtR ) xt + O( a ) , and hence (2.12) has the correct limit in a →
0; lim a → B = x/t . If we put (2.12) into (2.3), thenwe have A = ( rt ) { rtR − x − a + t − √ D } { (2 rtR − x + t (2 rtR − − a } . (2.14)The function A is indeed expressed by R and parameters r, t, a apart from I .We find that the second equation of (2.4) gives tB = Art − a (1 − rtR ) B , and if we use this equation, the quadratic equation (2.11) for B is written as x = Art − aB. (2.15)Now we put the expression (2.12) for B and the expression (2.14) for A into (2.15). Then we obtainthe following equation for R ,8 x ( rtR − n r t xR + 2 rt R − rtxR + ( x − a − t ) o × h r t x R − r t x { x + ( r − t } R + 2 rt [5 x + { (6 r − t − a } x + ( r − t ] R − [2 x + { (4 r − t − a } x + ( r − t { (2 r − t − a } ] i = 0 . (2.16)13e want to obtain R = R ( x ; r, t, a ) which solves (2.16) and satisfy the following continuitycondition with respect to a , lim a → R ( x ; r, t, a ) = R (2.17)with (2.8). This is given as a real solution of the cubic equation obtained from the last factor in(2.16), 8 r t x R − r t x { x + ( r − t } R + 2 rt [5 x + { (6 r − t − a } x + ( r − t ] R − [2 x + { (4 r − t − a } x + ( r − t { (2 r − t − a } ] = 0 . (2.18)Applying the Cardano formula, we obtain the solution as R ( x ; r, t, a ) = 2 x + ( r − t rtx − x + { a − (2 r + 1) t } x + ( r − t × / rtg / x − g / × / rtx , (2.19)where g = g ( x ; r, t, a ) is given by (1.18) with (1.15). Lemma 2.2
The solution (2.19) of (2.18) satisfies the continuity condition (2.17) with (2.8).Proof
If we set a = 0, (1.18) and (1.15) become g := g ( x ; r, t,
0) = − x + 3(2 r + 1) tx − h ( r − r + 1) t − p − S i x + 2( r − t , with S := S ( x ; r, t,
0) = t { x − r + 1) tx + ( r − t } . (2.20)In this case, the equality g = 14 (cid:26) x − ( r − t + 1 t p − S (cid:27) (2.21)is established. By putting (2.21) with (2.20) into (2.19) with a = 0, we can verify (2.17) with (2.8).We set ϕ = ϕ ( x ; r, t, a ) := 2 x { rtR ( x ; r, t, a ) − } . (2.22)Then it is easy to verify that the expression (2.19) for R is written as (1.19) for ϕ and that (2.14)gives 1 A − R = 1(2 rtx ) h t − a + √ D ) x − ϕ i , with D = − aϕ + ( t − a ) . (2.23)Therefore, if 2( t − a + √ D ) x − ϕ ≥ , (2.24)then (2.10) gives ρ ( x ) = q t − a + √ D ) x − ϕ πrtx . (2.25)14or the expression (1.20) for ρ ( x ) given in Theorem 1.1, we perform the further calculation asfollows. It is easy to verify the equality,2( t − a + √ D ) = (cid:16)p d − + p d + (cid:17) , where d ± are defined by (1.22). Therefore, we can see that2( t − a + √ D ) x − ϕ = n √ x (cid:16)p d − + p d + (cid:17) − ϕ o n √ x (cid:16)p d − + p d + (cid:17) + ϕ o = √ x + p d − + p d + ! − d d − √ x − p d − + p d + ! = √ x + p d − + p d + p d ! √ x + p d − + p d + − p d ! × p d + √ x − p d − + p d + ! p d − √ x + p d − + p d + ! = ( x − f L )( f R − x ) , where d is given by (1.22), and f L = f L ( x ; r, t, a ) and f R = f R ( x ; r, t, a ) are given by (1.21). Hence(2.25) is written as (1.20), provided that the condition (2.24) is equivalent with the condition x L ( r, t, a ) ≤ x ≤ x R ( r, t, a ) , (2.26)where x L ( r, t, a ) and x R ( r, t, a ) are defined by (1.17). Assume that t > , x >
0. Then if and only if the condition (2.24) is satisfied, ρ given by (2.25) ispositive or zero. And if and only if ρ ≥
0, its Hilbert transform
R/π given by the second equationof (2.1) and ϕ defined by (2.22) are real valued. By the explicit expression (1.19) with (1.18) for ϕ , the following is obvious. Lemma 2.3
If and only if S ( x ; r, t, a ) ≤ , ρ ( x ; r, t, a ) ≥ . Now we prove the following.
Lemma 2.4 If S ( x ; r, t, a ) = 0 , then ρ ( x ; r, t, a ) = 0 .Proof The formula (2.25) with (2.23) is written as ρ ( x ) = p ϕF ( ϕ )2 πrtx q a − t + √ D ) x + ϕ , (2.27)with F ( ϕ ) := 1 ϕ { a − t + √ D ) x + ϕ }{ t − a + √ D ) x − ϕ } = − ϕ + 4( t − a ) xϕ − ax . (2.28)15ence we consider the condition of F ( ϕ ) = 0. For ϕ defined by (2.22), the cubic equation (2.18) iswritten as H ( ϕ ) := ϕ + 2 { x − ( r − t } ϕ + [ x + { ( − r + 3) t − a } x + ( r − t ] ϕ + tx { x + ( r − a − t ) } = 0 . Therefore, F ( ϕ ) = 0 ⇐⇒ e F ( ϕ ) := F ( ϕ ) + H ( ϕ ) = 0 . Note that the cubic terms of ϕ are canceled and e F ( ϕ ) is reduced to be quadratic in ϕ . We obtain e F ( ϕ ) = 2 { x − ( r − t } ϕ + [ x + { ( − r + 7) t − a } x + ( r − t ] ϕ + x [( t − a ) x + ( r − t ( a − t )]= 2 { x − ( r − t } ( ϕ − ϕ − )( ϕ − ϕ + ) , where ϕ − = ϕ − ( x ; r, t, a ) = − x + { (7 − r ) t − a } x + ( r − t + √ ∆4 { x − ( r − t } ,ϕ + = ϕ + ( x ; r, t, a ) = − x + { (7 − r ) t − a } x + ( r − t − √ ∆4 { x − ( r − t } , (2.29)with ∆ = x + 2 {− (2 r − t + 59 a } x + [ { r (3 r −
8) + 35 } t − a (58 r − t + 25 a ] x − r − { (2 r − t + a } t x + ( r − t . The condition F ( ϕ ) = 0 is thus written as F ( ϕ + ) F ( ϕ − ) = 0. On the other hand, by (2.28) and(2.29), we can show that F ( ϕ + ) F ( ϕ − ) = H ( ϕ + ) H ( ϕ − )= − { x − { a + (12 r + 13) t } x + ( r − t { (12 r + 13) t − a } x − r − t } x { x − ( r − t } × S ( x ; r, t, a ) . Note that x − ( r − t >
0, if x >
0, since r ∈ (0 , , t ≥
0. Hence the statement of Lemma isconcluded.
Proof of Theorem 1.1
Assume that t >
0. By definition (1.17) of x L ( r, t, a ) and x R ( r, t, a ), andby Lemmas 2.3 and 2.4, we can conclude that the condition (2.24) is equivalent with the condition(2.26). Under the condition 2( t − a + √ D ) x − ϕ > x L ( r, t, a ) < x < x R ( r, t, a )), if ϕF ( ϕ ) = 0, then 2( a − t + √ D ) x + ϕ = 0 by the equality given by the first line of (2.28). Hence ρ ( x ) given by (2.27) is finite for x >
0. The proof of Theorem 1.1 is thus complete.
The constant term in the cubic function S ( x ; r, t, a ) of x given by (1.15) becomes 0 for arbitrary t > a ≥
0, if and only if r = 1. This implies Proposition 1.2 (i).16hen r = 1, the cubic equation (1.16) with (1.15) becomes x { ax − (8 a + 20 at − t ) x + 4( a − t ) } = 0 . For t > , a >
0, this equation has three real solutions; x = 0 and x = x ± := 18 a { a + 20 at − t ± p (8 a + 20 at − t ) − a ( a − t ) } = 18 a { a + 20 at − t ± √ t (8 a + t ) / } . When 0 < t < a , 8 a + 20 at − t > a + 20 at > √ t (8 a + t ) / = p (8 a + 20 at − t ) − a ( a − t ) < a + 20 at − t , and hence 0 < x − < x + . On the other hand, when t ≥ a , √ t (8 a + t ) / = p (8 a + 20 at − t ) − a ( a − t ) ≥ | a + 20 at − t | , and hence x − < < x + . Then we can conclude the following by the definitions of x L ( r, t, a ) and x R ( r, t, a ) given by (1.17). Lemma 2.5
Assume that a > . (i) When < t < a , x L (1 , t, a ) = 18 a { a + 20 at − t − √ t (8 a + t ) / } ,x R (1 , t, a ) = 18 a { a + 20 at − t + √ t (8 a + t ) / } , (2.30) and < x L (1 , t, a ) < x R (1 , t, a ) . (ii) When t ≥ a , x L (1 , t, a ) = 0 ,x R (1 , t, a ) = 18 a { a + 20 at − t + √ t (8 a + t ) / } > . Put t = a − ε, < ε ≪ x L given by (2.30). Then it is easy to verify that x L (1 , t, a ) = 427 ε a + 881 ε a + 52729 ε a + O( ε ) . (2.31)Hence Proposition 1.2 (ii) is proved. 17 .4 Proof of Proposition 1.3 In the case with r = 1, we have the following expressions from (1.15), (1.18), (1.19), and (2.23), S ( x ) := S ( x ; 1 , t, a ) = x [4 ax − (8 a + 20 at − t ) x + 4( a − t ) ] ,g ( x ) := g ( x ; 1 , t, a ) = x h − x + 9( t + 2 a ) x + 3 p − S ( x ) i ,ϕ ( x ) := ϕ ( x ; 1 , t, a ) = − x − / { x + 3( a − t ) } xg ( x ) / − g ( x ) / × / ,D ( x ) := − aϕ ( x ) + ( a − t ) . (2.32)Here we write ρ ( x ) := ρ ( x ; 1 , t, a ).First assume 0 < t < t c ( a ) = a . Put x L := x L (1 , t, a ) and let 0 < δ ≪
1. Since S ( x L ) = 0, wehave the expansions in the form, S ( x L + δ ) = c δ + c δ + O( δ ) ,g ( x L + δ ) = g ( x L ) + c δ / + c δ + O( δ / ) ,ϕ ( x L + δ ) = ϕ ( x L ) + c δ + O( δ / ) , where c j , j = 1 , . . . , t, a, x L , but independent of δ . It should be noted that, in theexpansion of ϕ ( x L + δ ), the coefficient of term δ / is proportional to d := { x L + 3( a − t ) } x L (cid:18) g ( x L ) (cid:19) / − (cid:18) g ( x L )2 (cid:19) / , where g ( x L ) = {− x L + 9( t + 2 a ) } x , and we can show that d ∝ S ( x L ) = 0. Then, if we note ρ ( x L ) = 0, (2.25) gives ρ ( x L + δ ) = δ / √ πtx L vuut t − a + p D ( x L ) − ax L p D ( x L ) + ϕ ( x L ) ! c + O( δ ) . (2.33)By (2.31) with ε := a − t , we see that ϕ ( x L ) = − a ε + O( ε ) ,D ( x L ) = 259 ε + O( ε ) ,c = − a ε − + O(1) . Hence by (2.33), Proposition 1.3 (i) is proved.Next assume t = t c ( a ) = a . Then (2.32) gives S ( x ) = a (4 x − a ) x = − a x + O( x ) ,g ( x ) = 2 × ax + O( x ) ,ϕ ( x ) = − a / x / + O( x ) ,D ( x ) = − aϕ ( x ) = 4 a / x / + O( x ) . t > t c ( a ) = a . Then (2.32) gives S ( x ) = − | ε | x + O( x ) ,g ( x ) = 6 √ | ε | / x / + O( x ) ,ϕ ( x ) = − t | ε | x + O( x / ) ,D ( x ) = | ε | + O( x ) , where | ε | = − ε = t − a . Then (2.33) proves Proposition 1.3 (iii). The proof of Proposition 1.3 ishence complete. It is obvious that the functions S ( x ; r, t, a ), g ( x ; r, t, a ), ϕ ( x ; r, t, a ), f L ( x ; r, t, a ), and f R ( x ; r, t, a ),which appeared in Theorem 1.1, are all homogeneous as multivariate functions of x, t, a for eachfixed value of r ∈ (0 , κ > S ( κx ; r, κt, κa ) = κ S ( x ; r, t, a ) ,g ( κx ; r, κt, κa ) = κ g ( x ; r, t, a ) , ϕ ( κx ; r, κt, κa ) = κϕ ( x ; r, t, a ) ,f L ( κx ; r, κt, κa ) = κf L ( x ; r, t, a ) , f R ( κx ; r, κt, κa ) = κf R ( x ; r, t, a ) . Then the scaling property of the three-parametric MP density (1.24) is concluded. If we set κ = 1 /t ,replace x by tx =: y , and take the limit t → ∞ for a fixed a >
0, Proposition 1.4 is proved.
In the present paper, we have studied the time-dependent complex Wishart ensemble of randommatrices with an external source. Following Blaizot, Nowak, and Warcho l [3], we have consideredthe hydrodynamic limit of the process of the squared-singular-values of random complex rectangularmatrices with a rectangularity r ∈ (0 , G ξ , which is equivalent with the nonlinear PDE for G ξ (1.8) under the initial distribution ξ ( dx ) = δ a ( dx ) , a >
0; a delta measure concentrated at x = a >
0. This algebraic equation (1.14)was given and its solution was studied by [3], but explicit expressions for the density function hasnot been available. In this paper we called the density function of this system the three-parametricMarcenko–Pastur (MP) density , ρ ( x ; r, t, a ) , r ∈ (0 , , t > , a ≥
0, and gave useful expressions to ρ ( x ; r, t, a ) (Theorem 1.1). As an application of the result, the dynamic critical phenomena wereclarified (Propositions 1.2 and 1.3), which are observed at the critical time t c ( a ) = a , if and only if r = 1 and a >
0. There we have introduced six kinds of critical exponents , ν = 3 , β = β = 12 , γ = 52 , γ = 1 δ = 13 , γ = 12 , which represent the singularities of the dynamic critical phenomena.The present results can be regarded as macroscopic descriptions of the system and the criticalphenomena. Microscopic descriptions have been also studied in several papers [16, 7, 11, 9] for thesimilar systems and the associated dynamic critical phenomena. Connection between these two19inds of descriptions [3] and the universality of such dynamic critical phenomena will be studiedin more detail in the future. As mentioned in Remark 3 given at the end of Section 1.3, in thecontext of high energy physics, the present macroscopic description can be regarded as a mean-field approximation for more precise theory of QCD which exhibits spontaneous chiral symmetrybreaking.As emphasized in [2, 3], the MP density of the Wishart random-matrix ensemble has been usedin a broad range of mathematical sciences, physics, and information theory (see the references in[2, 3]). It is expected that the non-centered Wishart ensembles/processes and the present three-parametric MP density will be also useful in many applications, where the mean zero conditioncannot be assumed.
Acknowledgements
The present authors thank Hiroya Baba for useful discussion when thepresent study was started. They also thank anonymous referees very much, who suggested themto discuss the present results in the context of the application of chGUE to QCD. This work wassupported by the Grant-in-Aid for Scientific Research (C) (No.19K03674), (B) (No.18H01124), and(S) (No.16H06338) of Japan Society for the Promotion of Science.
References [1] Anderson, G. W., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cam-bridge: Cambridge University Press, 2010[2] Blaizot, J.-P., Nowak, M. A., Warcho l, P.: Universal shocks in the Wishart random-matrixensemble. Phys. Rev. E , 052134/1–10 (2013)[3] Blaizot, J.-P., Nowak, M. A., Warcho l, P.: Universal shocks in the Wishart random-matrixensemble. II. Nontrivial initial conditions. Phys. Rev. E , 042130/1–7 (2014)[4] Br´ezin, E., Hikami, S.: Random Matrix Theory with an External Source. Springer Briefs inMathematical Physics, vol. 19, Springer, Singapore (2017)[5] Bru, M. F.: Wishart process. J. Theor. Probab. , 725–751 (1991)[6] Cabanal Duvillard, T., Guionnet, A.: Large deviations upper bounds for the laws of matrix-valued processes and non-communicative entropies. Ann. Probab. , 1205–1261 (2001)[7] Delvaux, S., Kuijlaars, A. B. J., Rom´an, P., Zhang, L.: Non-intersecting squared Bessel pathswith one positive starting and ending point. J. Anal. Math. no 1, 1250016 (2013)[12] Janik, R. A., Nowak, M. A., Papp, G., Zahed, I.: Critical scaling at zero virtuality in QCD.Phys. Lett. B , 9–14 (1999)[13] Katori, M., Tanemura, H.: Symmetry of matrix-valued stochastic processes and noncollidingdiffusion particle systems. J. Math. Phys. , 3058–3085 (2004)[14] Katori, M., Tanemura, H.: Noncolliding squared Bessel processes, J. Stat. Phys. , 592–615(2011)[15] K¨onig, W., O’Connell, N.: Eigenvalues of the Laguerre process as non-colliding squared Besselprocess. Electron. Commun. Probab. , 107–114 (2001)[16] Kuijlaars, A. B. J., Martinez-Finkelshtein, A., Wielonsky, F.: Non-intersecting squared Besselpaths: critical time and double scaling limit. Commun. Math. Phys. , 227–279 (2011)[17] Liu, Y., Warcho l, P., Zahed, I: Hydrodynamics of the Dirac spectrum. Phys. Lett. B ,303–307 (2016)[18] Marcenko, V. A., Pastur, L. A.: Distributions of eigenvalues for some sets of random matrices.Math. USSR-Sbornik1