1 Introduction and Main Results

1.1 Marcenko–Pastur law

Assume that \(M, N \in {\mathbb {N}}:=\{1,2, \dots \}\), \(M \ge N\). Consider \(M \times N\) complex random matrices \(K=(K_{jk})\) such that the real and the imaginary parts of elements are i.i.d. and normally distributed with mean \(\mu =0\) and variance \(\sigma ^2=1/2\). The normal distribution with mean \(\mu \) and variance \(\sigma ^2\) is denoted by \(N(\mu , \sigma ^2)\) and when a random variable X obeys \(N(\mu , \sigma ^2)\), we write it as \(X \sim N(\mu , \sigma ^2)\). Then the present setting is described as

$$\begin{aligned} \mathfrak {R}K_{jk} \sim N(0, 1/2), \quad \mathfrak {I}K_{jk} \sim N(0, 1/2), \quad j=1, \dots , M, \quad k=1, \dots , N. \end{aligned}$$

We consider a statistical ensemble of \(N \times N\) Hermitian random matrices L defined by

$$\begin{aligned} L= K^{\dagger } K, \end{aligned}$$
(1.1)

where \(K^{\dagger }\) 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 \(X_j^N, j=1, \dots , N\), which are nonnegative, since L is nonnegative definite by definition; \(X_j^N \in {\mathbb {R}}_{\ge 0}:= \{x \in {\mathbb {R}}: x \ge 0\}\). The positive square roots of them, \(\sqrt{X_j^N}, j=1, \dots , N\) are called singular values of random rectangular matrices K. In other words, the eigenvalue distribution of the Hermitian random matrices L can be regarded as the distribution of squares of singular values of the rectangular complex random matrices K in the complex Wishart random-matrix ensemble.

Let \({{\mathcal {C}}}_{\mathrm{c}}({\mathbb {R}})\) be the set of all continuous real-valued function with compact support on \({\mathbb {R}}\). We consider the empirical measure defined by

$$\begin{aligned} \Xi ^N(dx)= \frac{1}{N} \sum _{j=1}^N \delta _{X_j^N /M}(dx), \quad x \in {\mathbb {R}}_{\ge 0}, \end{aligned}$$
(1.2)

where \(\delta _y(dx)\) denotes a Dirac measure concentrated on y such that \(\int _{{\mathbb {R}}} f(x) \delta _y(dx)=f(y)\) for all \(f \in {{\mathcal {C}}}_{\mathrm{c}}({\mathbb {R}})\). Then we take the double limit \(N \rightarrow \infty \), \(M \rightarrow \infty \) for each fixed value of the rectangularity

$$\begin{aligned} r=\lim _{\begin{array}{c} N \rightarrow \infty , \\ M \rightarrow \infty \end{array}} \frac{N}{M} \in (0, 1]. \end{aligned}$$
(1.3)

We can prove that in this scaling limit (1.3), the empirical measure (1.2) converges weakly to a deterministic measure \(\rho (x) dx, x \in {\mathbb {R}}_{\ge 0}\) in the sense that \(\int _{{\mathbb {R}}_{\ge 0}} f(x) \Xi ^N(dx) \rightarrow \int _{{\mathbb {R}}_{\ge 0}} f(x) \rho (x) dx\) as \(N \rightarrow \infty \) for any \(f \in {{\mathcal {C}}}_{\mathrm{c}}({\mathbb {R}})\). Moreover, the probability density \(\rho \) in the limit measure has a finite support in \({\mathbb {R}}\) and it is explicitly given as a function of the parameter \( r \in (0, 1]\) as [18]

$$\begin{aligned} \rho (x; r) = \frac{\sqrt{(x-x_{\mathrm{L}}(r))(x_{\mathrm{R}}(r)-x)}}{2 \pi r x} \mathbf{1}_{(x_{\mathrm{L}}(r), x_{\mathrm{R}}(r))}(x) \end{aligned}$$
(1.4)

with

$$\begin{aligned} x_{\mathrm{L}}(r) :=(1-\sqrt{r})^2, \quad x_{\mathrm{R}}(r) :=(1+\sqrt{r})^2. \end{aligned}$$
(1.5)

Here \(\mathbf{1}_{\Lambda }(x)\), \(\Lambda \subset {\mathbb {R}}\) is an indicator function such that \(\mathbf{1}_{\Lambda }(x)=1\) if \(x \in \Lambda \), and \(\mathbf{1}_{\Lambda }(x)=0\) otherwise. This convergence theorem is known as the Marcenko–Pastur law for the Wishart random-matrix ensemble [1, 8, 18] and we call (1.4) the Marcenko–Pastur (MP) density in this paper.

1.2 Dynamical Extension of MP Density

A dynamical extension of the eigenvalue distribution of the Wishart random-matrix ensemble is realized by the solution \(\{ {X}^N_j(t) \in {\mathbb {R}}_{\ge 0}: t \ge 0, j=1, 2, \dots , N\}\) of the following system of stochastic differential equations (SDEs),

$$\begin{aligned} d {X}^N_j(t)&=2 \sqrt{ {X}^N_j(t)} dB_j(t) + 2 (\nu +1) dt \nonumber \\&\quad + 4 {X}^N_j(t) \sum _{\begin{array}{c} 1 \le k \le N, \\ k \not =j \end{array}} \frac{1}{ {X}^N_j(t)- {X}^N_k(t)} dt, \quad j=1, 2, \dots , N, \quad t \ge 0, \end{aligned}$$
(1.6)

where \(\nu =M-N\) and \(B_j(t), t \ge 0\) are independent one-dimensional standard Brownian motions starting from \(x^N_j \in {\mathbb {R}}_{\ge 0}\), \(j=1, \dots , N\). We assume that \(0 \le x^N_1 \le x^N_2 \le \cdots \le x^N_N < \infty \). This one-parameter family (\(\nu > 0\)) of N-particle stochastic processes was called (the eigenvalue process of) the Wishart process by Bru [5]. It is also called the Laguerre process or the noncolliding squared Bessel process [13,14,15].

We set \(\nu =(1-r) M =(1-r) N/r\), \(r \in (0, 1]\), and consider the empirical measure of the solution of SDEs (1.6),

$$\begin{aligned} \Xi ^N_t(dx) :=\frac{1}{N} \sum _{j=1}^N \delta _{ {X}^N_j(t)/M}(dx), \quad x \in {\mathbb {R}}_{\ge 0}, \quad t \ge 0. \end{aligned}$$
(1.7)

If the initial empirical measure satisfies some moment conditions and converges weakly to a measure, \(\Xi ^N_0(dx)=(1/N) \sum _{j=1}^N \delta _{x^N_j/M}(dx) \rightarrow \xi (dx)\) in the limit \(N \rightarrow \infty \), \(M \rightarrow \infty \) with \(r=N/M\) fixed in (0, 1], it is proved that \(\Xi ^N_t(dx)\) converges weakly to a time-dependent deterministic measure, which we denote here as \(\rho _{\xi }(x; r, t), t \ge 0\) [2, 6]. We define the Green’s function (the resolvent) \(G_{\xi }(z; r, t)\) by the Stieltjes transform of \(\rho _{\xi }\),

$$\begin{aligned} G_{\xi }(z; r, t) := \int _{{\mathbb {R}}} \frac{\rho _{\xi }(x; r, t)}{z-x} dx, \quad z \in {\mathbb {C}}\setminus {\mathbb {R}}. \end{aligned}$$

Then we can prove that this solves the following nonlinear partial differential equation (PDE) [2, 6],

$$\begin{aligned} \frac{\partial G_{\xi }}{\partial t} =-\frac{\partial G_{\xi }}{\partial z} + r \left\{ \frac{\partial G_{\xi }}{\partial z}-2 z G_{\xi } \frac{\partial G_{\xi }}{\partial z} - G_{\xi }^2 \right\} , \quad t \in [0, \infty ), \end{aligned}$$
(1.8)

under the initial condition, \(G_{\xi }(z; r, 0)=\int _{{\mathbb {R}}} \xi (dx)/(z-x)\), \(z \in {\mathbb {C}}\setminus {\mathbb {R}}\). Once the Green’s function \(G_{\xi }(z; r, 0), z \in {\mathbb {C}}\setminus {\mathbb {R}}\) is determined, we can obtain the density function following the Sokhotski–Plemelj theorem,

$$\begin{aligned} \rho _{\xi }(x; r, t) =- \mathfrak {I}\left[ \lim _{\varepsilon \rightarrow 0} \frac{1}{\pi } G_{\xi }(x+i \varepsilon ; r, t) \right] , \quad i :=\sqrt{-1}. \end{aligned}$$
(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 by setting the initial distribution,

$$\begin{aligned} \xi (dx)= \delta _0(dx) := \delta (x) dx, \end{aligned}$$
(1.10)

that is, all particles are concentrated on the origin; \(x^N_j=0, j \in {\mathbb {N}}\). By the method of complex characteristics, Blaizot et al. [2] showed that the Green’s function for (1.10), \(G_{\delta _0}(z)=G_{\delta _0}(z;r,t)\), is given by the solution of the equation

$$\begin{aligned} z=\frac{1}{G_{\delta _0}(z)} + \frac{t}{1-rt G_{\delta _0}(z)}, \quad z \in {\mathbb {C}}\setminus {\mathbb {R}}, \quad r \in (0, 1], \quad t \ge 0. \end{aligned}$$
(1.11)

They solved (1.11) and using the Sokhotski–Plemelj theorem (1.9) derived the time-dependent extension of (1.4),

$$\begin{aligned} \rho (x; r, t) := \rho _{\delta _0}(x; r, t) = \frac{\sqrt{(x-x_{\mathrm{L}}(r, t))(x_{\mathrm{R}}(r, t)-x)}}{2 \pi r t x} \mathbf{1}_{(x_{\mathrm{L}}(r, t), x_{\mathrm{R}}(r, t))}(x) \end{aligned}$$
(1.12)

with

$$\begin{aligned} x_{\mathrm{L}}(r, t) :=(1-\sqrt{r})^2t, \quad x_{\mathrm{R}}(r, t) :=(1+\sqrt{r})^2t, \quad t \in (0, \infty ). \end{aligned}$$
(1.13)

Since \(\rho (x; r, 1)\) is equal to the original MP density (1.4), the above provides a dynamical derivation of the Marcenko–Pastur law. The dependence on t of \(\rho (x; r, t)\) given by (1.12) with (1.13) is very simple, but we regard this as the two-parametric MP density in the present paper.

1.3 Main Results

Blaizot et al. [3] have studied the hydrodynamic limit of the Wishart eigenvalue-process starting from one-parameter family (\(a \ge 0\)) of the initial distribution,

$$\begin{aligned} \xi (dx)= \delta _a(dx) := \delta (x-a) dx. \end{aligned}$$

They showed that the Green’s function, \(G(z)= G_{\delta _a}(z; r, t)\), \(a \ge 0\), is obtained by the solution of the equation,

$$\begin{aligned} z= & {} \frac{1}{G_{\delta _a}(z)} +\frac{t}{1-r t G_{\delta _a}(z)}\nonumber \\&+\frac{a}{(1- r t G_{\delta _a}(z))^2}, \quad z \in {\mathbb {C}}\setminus {\mathbb {R}}, \quad r \in (0,1], \quad t \ge 0, \quad a \ge 0. \end{aligned}$$
(1.14)

They claimed in [3] that a proper solution of this equation yields \(\rho _{\delta _a}(x; r, t)\) via the Sokhotski–Plemelj theorem and showed an illustration (Fig.1 in [3]) of the time dependence of this density function for a special case with \(r=1\) and \(a=1\). The explicit formula of \(\rho _{\delta _a}(x; r, t)\) was, however, not given there. See also Section 8 in [17] and Section 3 in [10] for implicit expressions of \(G_{\delta _a}\).

We write the solution discussed in [3] as \(\rho (x; r, t, a)\) and call it the three-parametric Marcenko-Pastur (MP) density. The purpose of the present paper is to report useful expressions and detailed analysis of this density function \(\rho (x; r, t, a)\) on \({\mathbb {R}}_{\ge 0}\) with three parameters \(r \in (0, 1], t \ge 0\) and \(a \ge 0\).

The main theorem of the present paper is the following.

Theorem 1.1

Let

$$\begin{aligned} S(x; r, t, a)&:= 4 a x^3 - \{8 a^2+4a(3r+2)t-t^2\} x^2 \nonumber \\&\quad +2 [2 a^3-2a^2(5r-2) t + a\{r(6r-1)+1\} t^2 -(r+1)t^3 ] x \nonumber \\&\quad +(r-1)^2 t^2 \{a^2-a(4r-2)t+t^2\}. \end{aligned}$$
(1.15)

For \(r \in (0, 1], t >0, a \ge 0\), consider the case such that the cubic equation with respect to x,

$$\begin{aligned} S(x; r, t, a)=0, \end{aligned}$$
(1.16)

has three real solutions, \(x_1 \le x_2 \le x_3\), where \(x_j=x_j(r, t, a), j=1,2,3\). Define

$$\begin{aligned} x_{\mathrm{L}}(r, t, a) := x_2(r, t, a), \quad x_{\mathrm{R}}(r, t, a) :=x_3(r, t, a). \end{aligned}$$
(1.17)

Put

$$\begin{aligned} g&= g(x; r, t, a) \nonumber \\&:= -2 x^3+ 3 \{(2r+1)t+6a \} x^2 \nonumber \\&\quad -3 \Big [ (r-1)\{(2r+1)t-3a\} t - \sqrt{- 3 S} \Big ] x +2(r-1)^3 t^3, \end{aligned}$$
(1.18)

with \(S=S(x; r, t, a)\) given by (1.15), and define

$$\begin{aligned} \varphi&= \varphi (x; r, t, a) \nonumber \\&:= - \frac{2}{3} \{x-(r-1)t\} \nonumber \\&\quad -\frac{2^{1/3}}{3} \frac{x^2+\{3a-(2r+1)t\}x+t^2(r-1)^2}{g^{1/3}} -\frac{g^{1/3}}{3 \times 2^{1/3}}. \end{aligned}$$
(1.19)

Then the three-parametric MP density is given by

$$\begin{aligned} \rho (x; r, t, a) = \frac{\sqrt{(x-f_{\mathrm{L}}(x; r, t, a))(f_{\mathrm{R}}(x; r, t, a)-x)}}{2 \pi r x t} \mathbf{1}_{(x_{\mathrm{L}}(r, t, a), x_{\mathrm{R}}(r, t, a))}(x)\nonumber \\ \end{aligned}$$
(1.20)

with

$$\begin{aligned} f_{\mathrm{L}}(x; r, t, a):= & {} \left( \frac{\sqrt{d_-}+\sqrt{d_+}}{2} -\sqrt{d_0} \right) ^2, \quad \nonumber \\ f_{\mathrm{R}}(x; r, t, a):= & {} \left( \frac{\sqrt{d_-}+\sqrt{d_+}}{2} + \sqrt{d_0} \right) ^2, \end{aligned}$$
(1.21)

where

$$\begin{aligned} d_-&= t-a-2\sqrt{a \varphi }, \quad d_+ = t-a+2\sqrt{a \varphi }, \nonumber \\ d_0&= \varphi +x+\frac{t-a}{2} + \frac{1}{2} \sqrt{d_- d_+}. \end{aligned}$$
(1.22)

Remark 1

The formula (1.20) for the present three-parametric MP density seems to be similar to the original MP density (1.4) and the two-parametric MP density (1.12). We should note, however, that \(f_{\mathrm{L}}\) and \(f_{\mathrm{R}}\) appearing in (1.20) are not equal to the endpoints \(x_{\mathrm{L}}\) and \(x_{\mathrm{R}}\) of the support of density and they depend on x as shown by (1.21) with (1.18), (1.19) and (1.22). We can see that

$$\begin{aligned} \varphi (x; r, t, 0) := \lim _{a \rightarrow 0} \varphi (x; r, t, a)=-x+(r-1)t, \end{aligned}$$

and hence \(d_{\pm } \rightarrow t, d_0 \rightarrow rt\) as \(a \rightarrow 0\). Then as \(a \rightarrow 0\), \(f_{\mathrm{L}} \rightarrow x_{\mathrm{L}}(r; t)\), \(f_{\mathrm{R}} \rightarrow x_{\mathrm{R}}(r; t)\) with (1.13); that is, the dependence of \(f_{\mathrm{L}}\) and \(f_{\mathrm{R}}\) on x vanishes only in this limit. Theorem 1.1 states that for general \(a> 0\), the endpoints \(x_{\mathrm{L}}\) and \(x_{\mathrm{R}}\) of the support for the three-parametric MP density are given by the suitably chosen solutions (1.17) of the cubic equation (1.16) with (1.15) as proved in Sect. 2.2 below. That is, the formula (1.20) is universal, but the choice of solutions (1.17) depends on the parameters r, t, a. The equation (1.14) seems to be a simple perturbation of (1.11), but the solution turns out to have rich structures, by which we can describe dynamic critical phenomena at time \(t=t_{\mathrm{c}} :=a\) 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 theorem for the eigenvalue distribution of random matrix L given by (1.1) in the scaling limit (1.3), in which \(M \times N\) rectangular complex random matrices \(K=(K_{jk})\) are distributed as

$$\begin{aligned} \mathfrak {R}K_{kk}&\sim N(\sqrt{Ma}, t/2), \quad k=1, \dots , N, \nonumber \\ \mathfrak {R}K_{jk}&\sim N(0, t/2), \quad j=1, \dots , M, \quad k=1, \dots , N, \quad j \not =k, \nonumber \\ \mathfrak {I}K_{jk}&\sim N(0, t/2), \quad j=1, \dots , M, \quad k=1, \dots , N, \end{aligned}$$
(1.23)

\(t >0, r \in (0, 1]\). By (1.1), (1.23) gives

$$\begin{aligned} \mathbf{E}[L_{jk}]=M(a+t) \delta _{jk}, \quad j, k=1, \dots , N. \end{aligned}$$

When \(a > 0\), such an ensemble of random matrices will be called the Wishart ensemble with an external source or the non-centered Wishart ensemble, since even at \(t=0\), the diagonal elements of L have positive means, \(\mathbf{E}[L_{jj}]=Ma >0, j=1, \dots , N\), [4, 7, 9, 11, 16].

In Figure 1, we compare two histograms for the empirical measures (1.7) of the eigenvalues of matrices \(L=K^{\dagger } K\) given by K of size \(1000 \times 300\) (with the rectangularity \(r=300/1000=0.3\)), whose elements are following the probability law (1.23) with different parameters (t, a). When \((t, a)=(1, 0)\), the distribution of eigenvalues has a maximum at \(x \simeq 0.4\). When \((t, a)=(1, 1)\), due to an external source at \(x=a=1\), the distribution is shifted to the positive direction having a maximum at \(x \simeq 1\) and becomes broader. The former is well fitted by the original MP density \(\rho (x; r=0.3)\) and the latter is by the three-parametric MP density \(\rho (x; r=0.3, t=1, a=1)\) given by (1.20).

Fig. 1
figure 1

Two histograms for the empirical measures (1.7) with two different sets of parameters \((t, a)=(1,0)\) and (1, 1) are superposed in order to compare each other. They show the distributions of the eigenvalues of \(L=K^{\dagger } K\) given by K of size \(1000 \times 300\), whose elements are randomly generalized following the probability law (1.23) with \((t, a)=(1, 0)\) and (1, 1). The original MP density \(\rho (x; r=0.3)\) and the three-parametric MP density \(\rho (x; r=0.3, t=1, a=1)\) are shown by a thin curve and a thick curve, respectively. Due to an external source at \(x=a=1\), the eigenvalue distribution with \((t, a)=(1,1)\), which is well fitted by the three-parametric MP density \(\rho (x; r=0.3, t=1, a=1)\), is shifted to the positive direction and becomes broader compared with the original MP density \(\rho (x; r=0.3)\)

Full size image

For each values of rectangularity \(r \in (0, 1]\) and strength of an external source \(a \ge 0\), we can show time evolution of the support \((x_{\mathrm{L}}(r, t, a), x_{\mathrm{R}}(r, t, a) )\) of \(\rho (x; r, t, a)\) on the (x, t)-plane, \(({\mathbb {R}}_{\ge 0})^2\). Figure 2 shows the domains

$$\begin{aligned} {{\mathcal {D}}}(r, a) := \{ (x_{\mathrm{L}}(r, t, a), x_{\mathrm{R}}(r, t, a)) : t \ge 0\} \subset ({\mathbb {R}}_{\ge 0})^2 \end{aligned}$$

for \((r, a)=(0.3, 0)\) and \((r, a)=(0.3, 1)\). On the other hand, Figure 3 shows the domains for \((r, a)=(1, 0)\) and \((r, a)=(1, 1)\). As demonstrated by these figures, we can prove the following qualitative change of the domain when \(a>0\) and \(r=1\).

Fig. 2
figure 2

For \(r=0.3\), time evolution of the support \((x_{\mathrm{L}}, x_{\mathrm{R}})\) is shown on the (x, t)-plane for the two-parametric MP density \(\rho (x; r=0.3, t) := \rho (x; r=0.3, t, a=0)\) in the left, and for the three-parametric MP density with \(a=1\), \(\rho (x; r=0.3, t, a=1)\) in the right. The supports are extended in time, but the left edges of supports are kept to be positive, \(x_{\mathrm{L}} > 0\), for all \(t > 0\)

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Fig. 3
figure 3

For \(r=1\), time evolution of the support \((x_{\mathrm{L}}, x_{\mathrm{R}})\) is shown on the (x, t)-plane for the two-parametric MP density \(\rho (x; r=1, t) := \rho (x; r=1, t, a=0)\) in the left, and for the three-parametric MP density with \(a=1\), \(\rho (x; r=1, t, a=1)\) in the right. In the two-parametric MP density, the support starts from the singleton \(\{ 0 \}\) at \(t=0\) and the left edge of support \(x_{\mathrm{L}}\) is identically zero; \(x_{\mathrm{L}} \equiv 1\) for \(t \ge 0\). On the other hand, in the three-parametric MP density with \(a=1\), the support starts from the singleton \(\{ 1 \}\) at \(t=0\), and \(x_{\mathrm{L}} >0\) when \(t < t_{\mathrm{c}}=1\). As \(t \nearrow t_{\mathrm{c}}=1\), however, \(x_{\mathrm{L}} \searrow 0\) continuously, and then \(x_{\mathrm{L}} \equiv 1\) for \(t \ge t_{\mathrm{c}}=1\). We regard \(t_{\mathrm{c}}=1\) as a critical time

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Proposition 1.2

Assume that \(a > 0\).

  1. (i)

    If and only if \(r=1\), \({{\mathcal {D}}}(r,a)\) touches the origin \(x=0\). Otherwise, the left edge of \(\mathrm{supp} \, \rho (x; r, t, a)\) is strictly positive; \(x_{\mathrm{L}}(r, t, a) > 0\), \(r \in (0, 1)\).

  2. (ii)

       When \(r=1\), there is a critical time

    $$\begin{aligned} t_{\mathrm{c}}(a)=a \end{aligned}$$

    such that \(x_{\mathrm{L}}(1, t, a) >0\) while \(0 \le t < t_{\mathrm{c}}(a)\), and \(x_{\mathrm{L}}(1, t, a) \equiv 0\) for \(t \ge t_{\mathrm{c}}(a)\). In particular, just before the critical time \(t_{\mathrm{c}}(a)\), the left edge of \(\mathrm{supp} \, \rho (x; r, t, a)\) behaves as

    $$\begin{aligned} x_{\mathrm{L}}(1, t, a) \simeq \frac{4}{27 a^2}(t_{\mathrm{c}}(a)-t)^{\nu } \quad \text{ with } \nu =3 \quad \text{ as } t \nearrow t_{\mathrm{c}}(a). \end{aligned}$$

In the case with \(r=1\), the dynamic critical phenomena at the critical time \(t=t_{\mathrm{c}}(a)\) are observed in the vicinity of the origin as follows.

Proposition 1.3

When \(r=1\), the three-parametric MP density shows the following dynamic critical phenomena at \(t=t_{\mathrm{c}}(a)\).

  1. (i)

    For \(0< t < t_{\mathrm{c}}(a)\),

    $$\begin{aligned} \rho (x; 1, t, a) \simeq C_1(t, a) (x-x_{\mathrm{L}}(1, t, a))^{\beta _1} \quad \text{ with } \displaystyle {\beta _1 = \frac{1}{2}} \quad \text{ as } x \searrow x_{\mathrm{L}}(1, t, a), \end{aligned}$$

    where

    $$\begin{aligned} C_1(t, a) \simeq \frac{9 a}{4 \pi }(t_{\mathrm{c}}(a)-t)^{-\gamma _1} \quad \text{ with } \displaystyle {\gamma _1=\frac{5}{2}} \quad \text{ as } t \nearrow t_{\mathrm{c}}(a). \end{aligned}$$
  2. (ii)

       At \(t=t_{\mathrm{c}}(a)\),

    $$\begin{aligned} \rho (x; 1, t_{\mathrm{c}}(a), a) \simeq \frac{\sqrt{3}}{2 \pi } a^{-2/3} x^{-\gamma _2} \quad \text{ with } \displaystyle {\gamma _2 = \frac{1}{3}} \quad \text{ as } x \searrow 0. \end{aligned}$$
  3. (iii)

       For \(t > t_{\mathrm{c}}(a)\),

    $$\begin{aligned} \rho (x; 1, t, a) \simeq C_2(t, a) x^{-\gamma _3} \quad \text{ with } \displaystyle {\gamma _3 = \frac{1}{2}} \quad \text{ as } x \searrow 0, \end{aligned}$$

    where

    $$\begin{aligned} C_2(t, a) \simeq \frac{1}{\pi t_{\mathrm{c}}(a)} (t-t_{\mathrm{c}}(a))^{\beta _2} \quad \text{ with } \displaystyle {\beta _2 = \frac{1}{2}} \quad \text{ as } t \searrow t_{\mathrm{c}}(a). \end{aligned}$$

Remark 2

The critical exponents\(\nu =3\) and \(\gamma _2=1/3\) can be read in the argument given by [3]. Using the expressions given in Theorem 1.1 here we prove them as well as determining other critical exponents and critical amplitudes. The amplitude \(C_1\) of \(\rho \) in the subcritical time-region (\(0<t < t_{\mathrm{c}}(a)\)) diverges with the critical exponent \(\gamma _1=5/2\) as \(t \nearrow t_{\mathrm{c}}(a)\). Then both at the critical time (\(t=t_{\mathrm{c}}(a)\)) and in the supercritical time-region (\(t > t_{\mathrm{c}}(a)\)) \(\rho \) diverges as \(x \searrow 0\), but the critical exponents are different. In the supercritical time-region, the amplitude \(C_2\) of the diverging \(\rho \) with the critical exponent \(\gamma _3=1/2\) vanishes as \(t \searrow t_{\mathrm{c}}(a)\) (with the exponent \(\beta _2=1/2\)). Consequently the divergence of \(\rho \) as \(x \searrow 0\) is weakened at \(t=t_{\mathrm{c}}(a)\) having a smaller value of exponent as \(\gamma _2=1/3 < \gamma _3=1/2\). See Fig. 4. The proof of Proposition 1.3 given in Subsect. 2.4 implies the scaling relation

$$\begin{aligned} \nu =\beta _2+\gamma _1. \end{aligned}$$
Fig. 4
figure 4

Critical behavior of the three-parametric MP density \(\rho \) is shown for \(r=1\) and \(a=1\) with the critical time \(t_{\mathrm{c}}(1)=1\). The dashed curve denotes the emergence of \(\rho \) at \(x = x_{\mathrm{L}} \simeq 0.028\) with the critical exponent \(\beta _1=1/2\) at a subcritical time (\(t=0.5 t_{\mathrm{c}}(1)\)). The divergence of \(\rho \) as \(x \searrow 0\) at the critical time \(t=t_{\mathrm{c}}(1)\) is shown by a solid curve and that at a supercritical time (\(t=1.5 t_{\mathrm{c}}(1)\)) by a dotted curve. The former with the critical exponent \(\gamma _2=1/3\) is weaker than the latter with \(\gamma _3=1/2\)

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The functions S(x; r, t, a), g(x; r, t, a), \(\varphi (x; r, t, a)\), \(f_{\mathrm{L}}(x; r, t, a)\), and \(f_{\mathrm{R}}(x; r, t, a)\), which appear in Theorem 1.1, are all homogeneous as multivariate functions of x, t, a for each fixed value of \(r \in (0, 1]\). This fact implies the following scaling property of the three-parametric MP density,

$$\begin{aligned} \rho (\kappa x; r, \kappa t, \kappa a) =\frac{1}{\kappa } \rho (x; r, t, a), \quad r \in (0, 1], \end{aligned}$$
(1.24)

for an arbitrary parameter \(\kappa >0\). By this property, the following long-term behavior of the three-parametric MP density is readily concluded.

Proposition 1.4

For \(a \ge 0\),

$$\begin{aligned} \lim _{t \rightarrow \infty } \rho (y; r, t, a) dy \Big |_{y=tx} = \rho (x; r) dx, \end{aligned}$$

where \(\rho (x; r)\) 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 the original MP density by factor t. In this sense, the original Marcenko–Pastur law is universal and it describes the large-scale and long-term behavior of the Wishart ensemble and process.

Fig. 5
figure 5

Time evolution of the hydrodynamical density \(\rho ^{\mathrm{chiral}}(x; r, t, a)\) of the QCD Dirac operator in the critical case, which is obtained as (1.25) from the present three-parametric MP density with \(r=1\). For \(a=1\), \(\rho ^{\mathrm{chiral}}\) is plotted for \(t=0.5\) (the thinnest curve), 1.0, 1.5 and 3 (the thickest curve). When \(0 \le t \le t_{\mathrm{c}}=1\), the density at the origin \(\rho ^{\mathrm{chiral}}(0; r=1, t, a=1) =0\), while it becomes positive for \(t > t_{\mathrm{c}}\). For \(t > 3 t_{\mathrm{c}}\), \(\rho ^{\mathrm{chiral}}\) shows a relaxation in the sense of (1.27) to the universal density following Wigner’s semicircle law (1.28)

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Remark 3

So far we have studied the density \(\rho \) 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 \(\rho ^{\mathrm{chiral}}\) of the positive-signed and negative-signed singular values of random rectangular matrix K have been discussed [10, 12, 17]. For the transformation from \(\rho \) to \(\rho ^{\mathrm{chiral}}\), see Eq.(3.34) in [10], for instance. The present three-parametric MP density \(\rho (x; r, t, a)\) given by Theorem 1.1 provides the following hydrodynamical description of the time-depending spectrum for the QCD Dirac operator with parameters \(r \in (0, 1]\) and \(a \ge 0\),

$$\begin{aligned} \rho ^{\mathrm{chiral}}(x; r, t, a) =2 |x| \rho (x^2; r, t, a^2), \quad x \in {\mathbb {R}}, \quad t > 0, \end{aligned}$$
(1.25)

under the initial state

$$\begin{aligned} \rho ^{\mathrm{chiral}}(x; r, 0, a) =\delta (x+a)+\delta (x-a), \quad x \in {\mathbb {R}}. \end{aligned}$$

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 \(r=1\),

$$\begin{aligned} \rho ^{\mathrm{chiral}}(x; 1, t_{\mathrm{c}}(a), a)&\simeq \frac{\sqrt{3}}{\pi } a^{-4/3} |x|^{1/\delta } \quad \text{ with } \delta =3 \text{ as } |x| \rightarrow 0, \nonumber \\ \rho ^{\mathrm{chiral}}(0; 1, t, a)&\simeq \frac{2}{\pi t_{\mathrm{c}}(a^2)}(t-t_{\mathrm{c}}(a^2))^{\beta _2} \quad \text{ with } \displaystyle {\beta _2=\frac{1}{2}} \text{ as } t \searrow t_{\mathrm{c}}(a). \end{aligned}$$
(1.26)

Moreover, Proposition 1.4 implies through (1.25) that, when \(r=1\),

$$\begin{aligned} \lim _{t \rightarrow \infty } \rho ^{\mathrm{chiral}}(y; 1, t, a) dy \Big |_{y=\sqrt{t} x}&= \lim _{t \rightarrow \infty } \rho (t x^2; 1, t, a^2) d(t x^2)\nonumber \\&= \rho (x^2; 1) d x^2 = 2 x \rho (x^2; 1) dx\nonumber \\&= \rho ^{\mathrm{Wigner}}(x) dx, \quad x \in {\mathbb {R}}, \end{aligned}$$
(1.27)

where

$$\begin{aligned} \rho ^{\mathrm{Wigner}}(x)=\frac{1}{\pi } \sqrt{4-x^2} \mathbf{1}_{(-2, 2)}(x) \end{aligned}$$
(1.28)

is the density function describing Wigner’s semicircle law (see, for instance, [8]). As mentioned in [17], the time evolution of \(\rho ^{\mathrm{chiral}}\) from the two-peak shape with zero density at the origin \((0 \le t \le t_{\mathrm{c}}(a))\) to the universal shape \(\rho ^{\mathrm{Wigner}}\) after \(t \sim 3 t_{\mathrm{c}}(a)\) via a critical shape at \(t=t_{\mathrm{c}}(a)\) can be interpreted 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 with the density \(\rho ^{\mathrm{chiral}}(x; r, t, a)\) give a mean-field model for QCD and \(\delta =3\) and \(\beta _2=1/2\) in (1.26) are the mean-field values for the scaling exponents describing a condensation of light quarks to create massive constituents.

The paper is organized as follows. In Sect. 2 we give proofs of theorems and propositions given above. More precisely, Subsects. 2.1 and 2.2 are devoted to the proof of Theorem 1.1. The proofs of Propositions 1.2, 1.3, and 1.4 are given in Subsect. 2.3, 2.4, and 2.5, respectively. Concluding remarks are given in Sect. 3.

2 Proofs of Theorem and Propositions

2.1 Solving the Algebraic Equations for the Density and Its Hilbert Transform

For the Green’s function \(G_{\delta _a}(z)=G_{\delta _a}(z; r, t)\), \(a \ge 0\), we put

$$\begin{aligned} R&=R(x) := \lim _{\varepsilon \rightarrow 0} \mathfrak {R}G_{\delta _a}(x+i \varepsilon ),\\ I&=I(x) := - \lim _{\varepsilon \rightarrow 0} \mathfrak {I}G_{\delta _a}(x+i \varepsilon ), \end{aligned}$$

that is, \(\lim _{\varepsilon \rightarrow 0} G(x+i \varepsilon )=R(x) - i I(x)\). For the three-parametric MP density \(\rho (x) :=\rho (x; r, t,a)\), its Hilbert transform is defined by

The Sokhotski–Plemelj theorem states that

$$\begin{aligned} \rho (x)=\frac{I(x)}{\pi }, \quad {{\mathcal {H}}}[\rho (\cdot )](x) = \frac{R(x)}{\pi }. \end{aligned}$$
(2.1)

Let

$$\begin{aligned} A =\frac{1}{R^2+I^2}, \qquad B =\frac{1}{(1-rt R)^2+(rt I)^2}. \end{aligned}$$
(2.2)

By definition, we obtain the equation,

$$\begin{aligned} A=\frac{(rt)^2}{2 r t R-1 +1/B}. \end{aligned}$$
(2.3)

For the equation (1.14), we obtain the following.

Lemma 2.1

The equation (1.14) for the complex-valued function \(G_{\delta _a}(z), a \ge 0\) is equivalent with the following system of equations for the real-valued functions A and B,

$$\begin{aligned} x&=R A+ (1-rt R)t B + a \left[ (1-rt R)^2-(rt I)^2 \right] B^2,\nonumber \\ 0&=[A-rt^2B]-2a(1-rt R)rt B^2. \end{aligned}$$
(2.4)

Proof

We put \(z=x+i \varepsilon , x, \varepsilon \in {\mathbb {R}}\) in (1.14) and take the limit \(\varepsilon \rightarrow 0\). Then the real part and the imaginary part of the obtained equation give (2.4). \(\square \)

Before solving the system of equations (2.4) for general \(a \ge 0\), first we solve it for the special case \(a=0\). In this case (2.4) with (2.1) and (2.2) are simplified as

$$\begin{aligned} x&= \frac{R_0}{R_0^2+(\pi \rho _0)^2} +\frac{(1-r t R_0)t}{(1-rtR_0)^2+(rt)^2(\pi \rho _0)^2}\\&= \left\{ \frac{1}{R_0^2+(\pi \rho _0)^2}- \frac{rt^2}{(1-rtR_0)^2+(rt)^2(\pi \rho _0)^2} \right\} R_0 +\frac{t}{(1-rtR_0)^2+(rt)^2(\pi \rho _0)^2},\\ 0&=\frac{1}{R_0^2+(\pi \rho _0)^2}- \frac{rt^2}{(1-rtR_0)^2+(rt)^2(\pi \rho _0)^2}, \end{aligned}$$

for \(R_0 :=R(x; r,t,0)\) and \(\rho _0 :=\rho (x; r, t, 0)\). They give two different expressions for x,

$$\begin{aligned} x&= \frac{1}{rt \{R_0^2+(\pi \rho _0)^2\}}, \end{aligned}$$
(2.5)
$$\begin{aligned} x&= \frac{t}{(1-rtR_0)^2+(rt)^2(\pi \rho _0)^2}. \end{aligned}$$
(2.6)

From (2.5), we have the relation

$$\begin{aligned} (\pi \rho _0)^2=\frac{1}{r t x}- R_0^2. \end{aligned}$$
(2.7)

Combining this with (2.6), we have \(x=t/(1-2 r t R_0 + rt/x)\), which is solved as

$$\begin{aligned} R_0=\frac{x+(r-1)t}{2 rt x}. \end{aligned}$$
(2.8)

Put (2.8) into (2.7), we obtain

$$\begin{aligned} (\pi \rho _0)^2 = \frac{1}{(2 rt x)^2} \{-x^2+2(r+1)tx-(r-1)^2 t^2 \} = \frac{(x-x_{\mathrm{L}})(x_{\mathrm{R}}-x)}{(2 r t x)^2}, \end{aligned}$$
(2.9)

where \(x_{\mathrm{L}}=x_{\mathrm{L}}(r, t)\) and \(x_{\mathrm{R}}=x_{\mathrm{R}}(r, t)\) are given by (1.13) and the two-parametric MP density (1.12) is obtained as the positive square root of (2.9) for \(x_{\mathrm{L}} \le x \le x_{\mathrm{R}}\).

The above calculation suggests that it will be easier to obtain R than I. By (2.1) and the first equation of (2.2), if

$$\begin{aligned} \frac{1}{A} - R^2 \ge 0, \end{aligned}$$

then

$$\begin{aligned} \rho (x)=\frac{1}{\pi } \sqrt{ \frac{1}{A}-R^2}. \end{aligned}$$
(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 \(\rho \). Actually we will show that this strategy is successful in the following.

By eliminating A in (2.4), we obtain a quadratic equation for B as

$$\begin{aligned} 2a (1-rtR) B^2+(t-a)B-x=0. \end{aligned}$$
(2.11)

We choose the following solution of (2.11),

$$\begin{aligned} B=\frac{a-t+\sqrt{D}}{4a(1-rtR)}, \end{aligned}$$
(2.12)

with

$$\begin{aligned} D=D(x; r, t, a) =8a(1-rtR) x +(t-a)^2, \end{aligned}$$
(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

$$\begin{aligned} \sqrt{D} = -(a-t)+4a(1-rtR_0) \frac{x}{t}+\mathrm{O}(a^2), \end{aligned}$$

and hence (2.12) has the correct limit in \(a \rightarrow 0\); \(\lim _{a \rightarrow 0} B=x/t\). If we put (2.12) into (2.3), then we have

$$\begin{aligned} A=\frac{(rt)^2 \{2(2rt R-1)x-a+t-\sqrt{D}\} }{2\{(2rt R-1)^2x+t(2rt R-1)-a\}}. \end{aligned}$$
(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

$$\begin{aligned} tB=\frac{A}{rt}-2a(1-rtR)B^2, \end{aligned}$$

and if we use this equation, the quadratic equation (2.11) for B is written as

$$\begin{aligned} x=\frac{A}{rt}-aB. \end{aligned}$$
(2.15)

Now we put the expression (2.12) for B and the expression (2.14) for A into (2.15). Then we obtain the following equation for R,

$$\begin{aligned}&8 x (rt R-1) \Big \{ 4r^2t^2 xR^2 +2rt^2R -4rt x R + (x -a-t) \Big \} \nonumber \\&\quad \times \Big [ 8r^3 t^3 x^2 R^3 -8r^2t^2 x \{2x+(r-1)t \}R^2 \nonumber \\&\quad +2rt [ 5x^2+ \{(6r-5)t -a\} x+(r-1)^2t^2 ] R \nonumber \\&\quad -[ 2x^2+ \{ (4r -3)t -2a \} x +(r-1) t \{(2r-1)t-a\} ] \Big ] =0. \end{aligned}$$
(2.16)

We want to obtain \(R=R(x; r, t, a)\) which solves (2.16) and satisfy the following continuity condition with respect to a,

$$\begin{aligned} \lim _{a \rightarrow 0} R(x; r, t, a)=R_0 \end{aligned}$$
(2.17)

with (2.8). This is given as a real solution of the cubic equation obtained from the last factor in (2.16),

$$\begin{aligned}&8r^3 t^3 x^2 R^3 -8r^2t^2 x \{2x+(r-1)t \}R^2 \nonumber \\&\quad +2rt [ 5x^2+ \{(6r -5)t -a\} x+(r-1)^2t^2 ] R \nonumber \\&\quad -[ 2x^2+ \{(4r -3)t -2a\} x +(r-1) t \{(2r-1)t-a\} ] =0. \end{aligned}$$
(2.18)

Applying the Cardano formula, we obtain the solution as

$$\begin{aligned} R(x; r, t, a)= & {} \frac{2x+(r-1)t}{3rt x} \nonumber \\&- \frac{x^2 +\{ 3a -(2r+1)t \} x + (r-1)^2 t^2}{3 \times 2^{2/3}r t g^{1/3} x} -\frac{g^{1/3}}{6 \times 2^{1/3}rt x}, \end{aligned}$$
(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

$$\begin{aligned} g_0:= & {} g(x; r, t, 0) = -2 x^3+3(2r+1) t x^2-3 \Big [(r-1)(2r+1)t\\&-\sqrt{-3 S_0} \Big ] x + 2(r-1)^3 t^3, \end{aligned}$$

with

$$\begin{aligned} S_0 := S(x; r, t, 0)= t^2 \{x^2-2(r+1) t x+(r-1)^2 t^2\}. \end{aligned}$$
(2.20)

In this case, the equality

$$\begin{aligned} g_0=\frac{1}{4} \left\{ x -(r-1) t + \frac{1}{t} \sqrt{- 3 S_0} \right\} ^3 \end{aligned}$$
(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). \(\square \)

We set

$$\begin{aligned} \varphi =\varphi (x; r, t, a) :=2 x \{ r t R(x; r, t, a)-1\}. \end{aligned}$$
(2.22)

Then it is easy to verify that the expression (2.19) for R is written as (1.19) for \(\varphi \) and that (2.14) gives

$$\begin{aligned} \frac{1}{A}-R^2=\frac{1}{(2 r t x)^2} \Big [ 2(t-a+\sqrt{D})x -\varphi ^2 \Big ], \end{aligned}$$

with

$$\begin{aligned} D=-4 a \varphi + (t-a)^2. \end{aligned}$$
(2.23)

Therefore, if

$$\begin{aligned} 2(t-a+\sqrt{D})x -\varphi ^2 \ge 0, \end{aligned}$$
(2.24)

then (2.10) gives

$$\begin{aligned} \rho (x)=\frac{\sqrt{2(t-a+\sqrt{D})x -\varphi ^2}}{2 \pi r t x}. \end{aligned}$$
(2.25)

For the expression (1.20) for \(\rho (x)\) given in Theorem 1.1, we perform the further calculation as follows. It is easy to verify the equality,

$$\begin{aligned} 2(t-a+\sqrt{D})=\left( \sqrt{d_-}+\sqrt{d_+}\right) ^2, \end{aligned}$$

where \(d_{\pm }\) are defined by (1.22). Therefore, we can see that

$$\begin{aligned}&2(t-a+\sqrt{D}) x-\varphi ^2\nonumber \\&\quad =\left\{ \sqrt{x} \left( \sqrt{d_-}+\sqrt{d_+}\right) -\varphi \right\} \left\{ \sqrt{x} \left( \sqrt{d_-}+\sqrt{d_+}\right) + \varphi \right\} \nonumber \\&\quad =\left\{ \left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right) ^2-d_0 \right\} \left\{ d_0 - \left( \sqrt{x}-\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right) ^2 \right\} \\&\quad = \left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} + \sqrt{d_0} \right) \left( \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} - \sqrt{d_0} \right) \\&\qquad \times \left( \sqrt{d_0} + \sqrt{x}-\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right) \left( \sqrt{d_0} - \sqrt{x}+\frac{\sqrt{d_-} + \sqrt{d_+}}{2} \right) \\&\quad =(x-f_{\mathrm{L}})(f_{\mathrm{R}}-x), \end{aligned}$$

where \(d_0\) is given by (1.22), and \(f_{\mathrm{L}}=f_{\mathrm{L}}(x; r, t, a)\) and \(f_{\mathrm{R}}=f_{\mathrm{R}}(x; r, t, a)\) are given by (1.21). Hence (2.25) is written as (1.20), provided that the condition (2.24) is equivalent with the condition

$$\begin{aligned} x_{\mathrm{L}}(r, t, a) \le x \le x_{\mathrm{R}}(r, t, a), \end{aligned}$$
(2.26)

where \(x_{\mathrm{L}}(r, t, a)\) and \(x_{\mathrm{R}}(r, t, a)\) are defined by (1.17).

2.2 Determining the Support of Density Function

Assume that \(t>0, x >0\). Then if and only if the condition (2.24) is satisfied, \(\rho \) given by (2.25) is positive or zero. And if and only if \(\rho \ge 0\), its Hilbert transform \(R/\pi \) given by the second equation of (2.1) and \(\varphi \) defined by (2.22) are real valued. By the explicit expression (1.19) with (1.18) for \(\varphi \), the following is obvious.

Lemma 2.3

If and only if \(S(x; r, t, a) \le 0\), \(\rho (x; r, t, a) \ge 0\).

Now we prove the following.

Lemma 2.4

If \(S(x; r, t, a)=0\), then \(\rho (x; r, t, a)=0\).

Proof

The formula (2.25) with (2.23) is written as

$$\begin{aligned} \rho (x)=\frac{\sqrt{\varphi F(\varphi )}}{2 \pi r t x \sqrt{2(a-t+\sqrt{D})x + \varphi ^2}}, \end{aligned}$$
(2.27)

with

$$\begin{aligned} F(\varphi )&:= \frac{1}{\varphi } \{2(a-t+\sqrt{D})x+\varphi ^2 \} \{2(t-a+\sqrt{D})x-\varphi ^2\}\nonumber \\&= -\varphi ^3+4(t-a)x \varphi -16a x^2. \end{aligned}$$
(2.28)

Hence we consider the condition of \(F(\varphi )=0\). For \(\varphi \) defined by (2.22), the cubic equation (2.18) is written as

$$\begin{aligned} H(\varphi )&:= \varphi ^3 + 2 \{x-(r-1)t \} \varphi ^2 + [ x^2+\{(-2r+3)t-a\} x +(r-1)^2 t^2 ] \varphi \\&\quad +t x \{x + (r-1)(a-t) \} =0. \end{aligned}$$

Therefore,

$$\begin{aligned} F(\varphi )=0 \quad \Longleftrightarrow \quad {\widetilde{F}}(\varphi ) := F(\varphi )+H(\varphi ) = 0. \end{aligned}$$

Note that the cubic terms of \(\varphi \) are canceled and \({\widetilde{F}}(\varphi )\) is reduced to be quadratic in \(\varphi \). We obtain

$$\begin{aligned} {\widetilde{F}}(\varphi )&= 2 \{x-(r-1)t \} \varphi ^2 +[ x^2 +\{(-2r+7) t - 5a\} x +(r-1)^2 t^2 ] \varphi \\&\qquad + x [(t-16a)x+(r-1)t(a-t)]\\&\quad =2 \{x-(r-1)t \} (\varphi -\varphi _-)(\varphi -\varphi _+), \end{aligned}$$

where

$$\begin{aligned} \varphi _-&= \varphi _-(x; r, t, a)= -\frac{x^2+\{(7-2r)t-5a\} x+(r-1)^2 t^2 + \sqrt{\Delta }}{4\{x-(r-1)t\}},\nonumber \\ \varphi _+&= \varphi _+(x; r, t, a)= -\frac{x^2+\{(7-2r)t-5a\} x+(r-1)^2 t^2 - \sqrt{\Delta }}{4\{x-(r-1)t\}}, \end{aligned}$$
(2.29)

with

$$\begin{aligned} \Delta&=x^4 + 2 \{-(2r-3)t + 59 a \} x^3 +[ \{2r(3r-8)+35\} t^2-2a(58r-33) t + 25 a^2 ] x^2\\&\quad -2 (r-1)^2 \{(2r-3) t + a \} t^2 x + (r-1)^4 t^4. \end{aligned}$$

The condition \(F(\varphi )=0\) is thus written as \(F(\varphi _+) F(\varphi _-)=0\). On the other hand, by (2.28) and (2.29), we can show that

$$\begin{aligned} F(\varphi _+) F(\varphi _-)&= H(\varphi _+) H(\varphi _-) \\&= -\frac{ \{4 x^3 - \{100 a+(12r+13)t\} x^2 + (r-1) t \{(12r+13) t-25 a\} x - 4 (r-1)^3 t^3\} x^2 }{8\{x-(r-1)t\}^3} \\&\quad \times S(x; r, t, a). \end{aligned}$$

Note that \(x-(r-1)t >0\), if \(x>0\), since \(r \in (0, 1], t \ge 0\). Hence the statement of Lemma is concluded. \(\square \)

Proof of Theorem 1.1

Assume that \(t > 0\). By definition (1.17) of \(x_{\mathrm{L}}(r, t, a)\) and \(x_{\mathrm{R}}(r, t, a)\), and by Lemmas 2.3 and 2.4, we can conclude that the condition (2.24) is equivalent with the condition (2.26). Under the condition \(2(t-a+\sqrt{D})x-\varphi ^2>0\) (equivalently, \(x_{\mathrm{L}}(r, t, a)< x < x_{\mathrm{R}}(r, t, a)\)), if \(\varphi F(\varphi ) \not =0\), then \(2(a-t+\sqrt{D})x+\varphi ^2 \not =0\) by the equality given by the first line of (2.28). Hence \(\rho (x)\) given by (2.27) is finite for \(x>0\). The proof of Theorem 1.1 is thus complete. \(\square \)

2.3 Proof of Proposition 1.2

The constant term in the cubic function S(x; r, t, a) of x given by (1.15) becomes 0 for arbitrary \(t > 0\) and \(a \ge 0\), if and only if \(r=1\). This implies Proposition 1.2 (i).

When \(r=1\), the cubic equation (1.16) with (1.15) becomes

$$\begin{aligned} x \{ 4 a x^2- (8 a^2+20 a t - t^2) x + 4(a-t)^3 \} =0. \end{aligned}$$

For \(t>0, a>0\), this equation has three real solutions; \(x=0\) and

$$\begin{aligned} x =x_{\pm }&:= \frac{1}{8a} \{ 8a^2+20at-t^2 \pm \sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3} \}\\&= \frac{1}{8a} \{ 8a^2+20at-t^2 \pm \sqrt{t} (8a+t)^{3/2} \}. \end{aligned}$$

When \(0< t < a\), \(8a^2+20at-t^2> 7a^2+20at>0\),

$$\begin{aligned} \sqrt{t} (8a+t)^{3/2} =\sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3} < 8a^2+20at-t^2, \end{aligned}$$

and hence \(0< x_- < x_+\). On the other hand, when \(t \ge a\),

$$\begin{aligned} \sqrt{t} (8a+t)^{3/2} =\sqrt{(8a^2+20at-t^2)^2-64a(a-t)^3} \ge |8a^2+20at-t^2|, \end{aligned}$$

and hence \(x_-< 0 < x_+\). Then we can conclude the following by the definitions of \(x_{\mathrm{L}}(r, t, a)\) and \(x_{\mathrm{R}}(r, t, a)\) given by (1.17).

Lemma 2.5

Assume that \(a > 0\).

  1. (i)

    When \(0< t < a\),

    $$\begin{aligned} x_{\mathrm{L}}(1, t, a)&= \frac{1}{8 a} \{ 8 a^2 + 20 a t - t^2 -\sqrt{t} (8a+t)^{3/2} \}, \nonumber \\ x_{\mathrm{R}}(1, t, a)&= \frac{1}{8 a} \{ 8 a^2 + 20 a t - t^2 +\sqrt{t} (8a+t)^{3/2} \}, \end{aligned}$$
    (2.30)

    and

    $$\begin{aligned} 0< x_{\mathrm{L}}(1, t, a) < x_{\mathrm{R}}(1, t, a). \end{aligned}$$
  2. (ii)

    When \(t \ge a\),

    $$\begin{aligned} x_{\mathrm{L}}(1, t, a)&= 0, \\ x_{\mathrm{R}}(1, t, a)&= \frac{1}{8 a} \{ 8 a^2 + 20 a t - t^2 +\sqrt{t} (8a+t)^{3/2} \} > 0. \end{aligned}$$

Put \(t=a-\varepsilon , 0 < \varepsilon \ll 1\) in \(x_{\mathrm{L}} \) given by (2.30). Then it is easy to verify that

$$\begin{aligned} x_{\mathrm{L}}(1, t, a)=\frac{4}{27} \frac{\varepsilon ^3}{a^2} + \frac{8}{81} \frac{\varepsilon ^4}{a^3} +\frac{52}{729} \frac{\varepsilon ^5}{a^4} +\mathrm{O}(\varepsilon ^6). \end{aligned}$$
(2.31)

Hence Proposition 1.2 (ii) is proved.

2.4 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),

$$\begin{aligned} S_1(x)&:= S(x; 1, t, a) =x [ 4ax^2-(8a^2+20 a t-t^2) x +4(a-t)^3],\nonumber \\ g_1(x)&:= g(x; 1, t, a) = x \left[ -2 x^2+9(t+2a)x + 3 \sqrt{-3 S_1(x)} \right] ,\nonumber \\ \varphi _1(x)&:= \varphi (x; 1, t, a) = -\frac{2}{3} x - \frac{2^{1/3}}{3} \frac{\{x+3(a-t)\}x}{g_1(x)^{1/3}} -\frac{g_1(x)^{1/3}}{3 \times 2^{1/3}},\nonumber \\ D_1(x)&:= - 4 a \varphi _1(x)+(a-t)^2. \end{aligned}$$
(2.32)

Here we write \(\rho _1(x):= \rho (x; 1, t, a)\).

First assume \(0< t < t_{\mathrm{c}}(a)=a\). Put \(x_{\mathrm{L}} := x_{\mathrm{L}}(1, t, a)\) and let \(0 < \delta \ll 1\). Since \(S_1(x_L)=0\), we have the expansions in the form,

$$\begin{aligned} S_1(x_{\mathrm{L}}+\delta )&= c_1 \delta + c_2 \delta ^2 + \mathrm{O}(\delta ^3),\\ g_1(x_{\mathrm{L}}+\delta )&= g_1(x_{\mathrm{L}})+c_3 \delta ^{1/2}+c_4 \delta + \mathrm{O}(\delta ^{3/2}),\\ \varphi _1(x_{\mathrm{L}}+\delta )&= \varphi _1(x_{\mathrm{L}})+ c_5 \delta + \mathrm{O}(\delta ^{3/2}), \end{aligned}$$

where \(c_j, j=1, \dots , 5\) are functions of \(t, a, x_{\mathrm{L}}\), but independent of \(\delta \). It should be noted that, in the expansion of \(\varphi _1(x_{\mathrm{L}}+\delta )\), the coefficient of term \(\delta ^{1/2}\) is proportional to

$$\begin{aligned} d:= \{x_{\mathrm{L}}+3(a-t)\}x_{\mathrm{L}} \left( \frac{2}{g_1(x_{\mathrm{L}})} \right) ^{1/3} -\left( \frac{g_1(x_{\mathrm{L}})}{2} \right) ^{1/3}, \end{aligned}$$

where \(g_1(x_{\mathrm{L}})=\{-2 x_{\mathrm{L}}+9(t+2a)\}x_{\mathrm{L}}^{2}\), and we can show that \(d \propto S_1(x_{\mathrm{L}})=0\). Then, if we note \(\rho _1(x_{\mathrm{L}})=0\), (2.25) gives

$$\begin{aligned} \rho _1(x_{\mathrm{L}}+\delta ) =\frac{\delta ^{1/2}}{\sqrt{2} \pi t x_{\mathrm{L}}} \sqrt{ t-a+\sqrt{D_1(x_{\mathrm{L}})} -\left( \frac{2a x_{\mathrm{L}}}{\sqrt{D_1(x_{\mathrm{L}})}} +\varphi _1(x_{\mathrm{L}}) \right) c_5} +\mathrm{O}(\delta ).\nonumber \\ \end{aligned}$$
(2.33)

By (2.31) with \(\varepsilon := a-t\), we see that

$$\begin{aligned} \varphi _1(x_{\mathrm{L}})&= - \frac{4}{9 a} \varepsilon ^2 + \mathrm{O}(\varepsilon ^3),\\ D_1(x_{\mathrm{L}})&= \frac{25}{9} \varepsilon ^2 + \mathrm{O}(\varepsilon ^3),\\ c_5&= - \frac{5a}{3} \varepsilon ^{-1} + \mathrm{O}(1). \end{aligned}$$

Hence by (2.33), Proposition 1.3 (i) is proved.

Next assume \(t=t_{\mathrm{c}}(a)=a\). Then (2.32) gives

$$\begin{aligned} S_1(x)&= a(4x-27a) x^2 =-27 a^2 x^2+\mathrm{O}(x^3\\ g_1(x)&= 2 \times 3^3 a x^2 + \mathrm{O}(x^3),\\ \varphi _1(x)&= - a^{1/3} x^{2/3} + \mathrm{O}(x),\\ D_1(x)&= - 4 a \varphi _1(x) = 4 a^{4/3} x^{2/3} +\mathrm{O}(x). \end{aligned}$$

Then (2.33) proves Proposition 1.3 (ii).

Finally assume \(t > t_{\mathrm{c}}(a)=a\). Then (2.32) gives

$$\begin{aligned} S_1(x)&= - 4 |\varepsilon |^3 x+\mathrm{O}(x^2),\\ g_1(x)&= 6 \sqrt{3} |\varepsilon |^{3/2} x^{3/2} + \mathrm{O}(x^2),\\ \varphi _1(x)&= -\frac{t}{|\varepsilon |} x + \mathrm{O}(x^{3/2}),\\ D_1(x)&= |\varepsilon |^2 + \mathrm{O}(x), \end{aligned}$$

where \(|\varepsilon |=-\varepsilon =t-a\). Then (2.33) proves Proposition 1.3 (iii). The proof of Proposition 1.3 is hence complete.

2.5 Long-Term Scaling and Proof of Proposition 1.4

It is obvious that the functions S(x; r, t, a), g(x; r, t, a), \(\varphi (x; r, t, a)\), \(f_{\mathrm{L}}(x; r, t, a)\), and \(f_{\mathrm{R}}(x; r, t, a)\), which appeared in Theorem 1.1, are all homogeneous as multivariate functions of x, t, a for each fixed value of \(r \in (0, 1]\). Actually, we see that, for an arbitrary parameter \(\kappa >0\),

$$\begin{aligned}&S(\kappa x; r, \kappa t, \kappa a) = \kappa ^4 S(x; r, t, a), \quad \\&g(\kappa x; r, \kappa t, \kappa a) = \kappa ^3 g(x; r, t, a), \quad \varphi (\kappa x; r, \kappa t, \kappa a) = \kappa \varphi (x; r, t, a),\\&f_{\mathrm{L}}(\kappa x; r, \kappa t, \kappa a) = \kappa f_{\mathrm{L}}(x; r, t, a), \quad f_{\mathrm{R}}(\kappa x; r, \kappa t, \kappa a) = \kappa f_{\mathrm{R}}(x; r, t, a). \end{aligned}$$

Then the scaling property of the three-parametric MP density (1.24) is concluded. If we set \(\kappa =1/t\), replace x by \(t x =: y\), and take the limit \(t \rightarrow \infty \) for a fixed \(a >0\), Proposition 1.4 is proved.

3 Concluding Remarks

In the present paper, we have studied the time-dependent complex Wishart ensemble of random matrices with an external source. Following Blaizot et al. [3], we have considered the hydrodynamic limit of the process of the squared-singular-values of random complex rectangular matrices with a rectangularity \(r \in (0, 1]\). We solved the algebraic equation (1.14) for the Green’s function \(G_{\xi }\), which is equivalent with the nonlinear PDE for \(G_{\xi }\) (1.8) under the initial distribution \(\xi (dx)=\delta _a(dx), a > 0\); 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 has not been available. In this paper we called the density function of this system the three-parametric Marcenko–Pastur (MP) density, \(\rho (x; r, t, a), r \in (0, 1], t >0, a \ge 0\), and gave useful expressions to \(\rho (x; r, t, a)\) (Theorem 1.1). As an application of the result, the dynamic critical phenomena were clarified (Propositions 1.2 and 1.3), which are observed at the critical time \(t_{\mathrm{c}}(a)=a\), if and only if \(r=1\) and \(a>0\). There we have introduced six kinds of critical exponents,

$$\begin{aligned} \nu =3, \quad \beta _1=\beta _2=\frac{1}{2}, \quad \gamma _1=\frac{5}{2}, \quad \gamma _2 =\frac{1}{\delta }=\frac{1}{3}, \quad \gamma _3=\frac{1}{2}, \end{aligned}$$

which represent the singularities of the dynamic critical phenomena.

The present results can be regarded as macroscopic descriptions of the system and the critical phenomena. Microscopic descriptions have been also studied in several papers [7, 9, 11, 16] for the similar systems and the associated dynamic critical phenomena. Connection between these two kinds of descriptions [3] and the universality of such dynamic critical phenomena will be studied in more detail in the future. As mentioned in Remark 3 given at the end of Sect. 1.3, in the context 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 symmetry breaking.

As emphasized in [2, 3], the MP density of the Wishart random-matrix ensemble has been used in 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 condition cannot be assumed.