The Interpolating Airy Kernels for the $\beta =1$ and $\beta =4$ Elliptic Ginibre Ensembles

Abstract

We consider two families of non-Hermitian Gaussian random matrices, namely the elliptic Ginibre ensembles of asymmetric $N$-by-$N$ matrices with Dyson index $\beta =1$ (real elements) and with $\beta =4$ (quaternion-real elements). Both ensembles have already been solved for finite $N$ using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large-$N$ limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the “interpolating” Airy kernels, since we can recover—as opposing limiting cases—not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for $\beta =2$, which we rederive here as well, this completes the analysis of all three elliptic Ginibre ensembles in the microscopic scaling limit at the spectral edge.

Appendices

Appendix: 1 The Deformed Airy Function

In this appendix we introduce the deformed Airy function

$$\begin{aligned} {{\mathrm{Aid}}}(Z,\sigma ) \equiv \exp \left[ \frac{\sigma ^6}{12} + \frac{\sigma ^2 Z}{2} \right] {{\mathrm{Ai}}}\left( Z + \frac{\sigma ^4}{4} \right) , \end{aligned}$$

(7.1)

and consider three of its scaling limits. Using the large argument asymptotic behaviour of the Airy function (see Eq. 10.4.59 of [47])

$$\begin{aligned} {{\mathrm{Ai}}}(w)&\sim \frac{\exp \left[ {} - \tfrac{2}{3} w^{3/2} \right] }{2\sqrt{\pi }w^{1/4}} \left( 1 - \frac{5}{48w^{3/2}} + \mathcal {O}(w^{-3}) \right) \nonumber \\&= \frac{1}{2\sqrt{\pi }} \, \exp \left[ {} - \tfrac{2}{3} w^{3/2} - \tfrac{1}{4} \log w + \log \left( 1 - \frac{5}{48w^{3/2}} + \mathcal {O}(w^{-3}) \right) \right] \end{aligned}$$

(7.2)

we can easily show that, at large $\sigma$ and fixed $Z$ (or, at least, for $Z$ growing more slowly than $\sigma ^4$), we have

$$\begin{aligned} {{\mathrm{Aid}}}(Z,\sigma ) = \frac{1}{\sqrt{2\pi }\,\sigma } \exp \left[ {} - \, \frac{Z^2}{2\sigma ^2} - \frac{Z}{\sigma ^4} + \frac{2Z^3-5}{6\sigma ^6} + \cdots \right] . \end{aligned}$$

(7.3)

From this, we immediately have the first of our scaling limits,

$$\begin{aligned} \lim _{\sigma \rightarrow \infty } \sigma \, {{\mathrm{Aid}}}(\sigma u, \sigma ) = \frac{1}{\sqrt{2\pi }} \, e^{-u^2/2} \end{aligned}$$

(7.4)

where we scale the first argument proportional to $\sigma$.

We now consider a second scaling limit. In our interpolating Airy kernels, e.g. Eqs. (3.7) and (4.21), the deformed Airy function typically appears in combination with an additional factor:

$$\begin{aligned} f(Z,t,\sigma ) \equiv \exp \left[ - \frac{Y^2}{2\sigma ^2} \right] {{\mathrm{Aid}}}(Z+t, \sigma ). \end{aligned}$$

(7.5)

Consider therefore the behaviour of

$$\begin{aligned} g(Z,t,\sigma )&\equiv f(X + i\sigma Y, t, \sigma ) \nonumber \\&= \exp \left[ - \, \frac{Y^2}{2} \right] \, {{\mathrm{Aid}}}(X + i\sigma Y + t, \sigma ), \end{aligned}$$

(7.6)

where it should be noted that we are scaling the real and imaginary parts of the argument $Z$ differently. At large $\sigma$ and fixed $X$ and $Y$ (or, at least, with $X$ and $Y$ growing sufficiently slowly with $\sigma$), we then have from Eq. (7.3)

$$\begin{aligned}&g(Z,t,\sigma )\nonumber \\&\quad = \frac{1}{\sqrt{2\pi }\,\sigma } \, \exp \left[ - \frac{Y^2}{2} \right] \, \exp \left[ - \frac{(X + i\sigma Y + t)^2}{2\sigma ^2} - \frac{X + i\sigma Y + t}{\sigma ^4} + \frac{2(X + i\sigma Y + t)^3-5}{6\sigma ^6} + \ldots \right] \nonumber \\&\quad = \exp \left[ - \frac{iXY}{\sigma } - \frac{iY^3}{3\sigma ^3} \right] \, h(Z,t,\sigma ) \end{aligned}$$

(7.7)

where

$$\begin{aligned} h(Z,t,\sigma )&= \frac{1}{\sqrt{2\pi }\,\sigma } \, \exp \left[ \left( {} - \frac{X^2}{2\sigma ^2} - \frac{iY}{\sigma ^3} - \frac{X}{\sigma ^4} - \frac{Y^2X}{\sigma ^4} + \frac{iX^2Y}{\sigma ^5} + \frac{X^3}{3\sigma ^6} - \frac{5}{6\sigma ^6} + \ldots \right) \right. \nonumber \\&\qquad \qquad \qquad {} - t \left( \frac{iY}{\sigma } + \frac{X}{\sigma ^2} + \frac{1}{\sigma ^4} + \frac{Y^2}{\sigma ^4} - \frac{2iXY}{\sigma ^5} - \frac{X^2}{\sigma ^6} + \ldots \right) \nonumber \\&\qquad \qquad \qquad \qquad \left. {} - t^2 \left( \frac{1}{2\sigma ^2} - \frac{iY}{\sigma ^5} - \frac{X}{\sigma ^6} + \ldots \right) + \frac{t^3}{3\sigma ^6} + \ldots \right] . \end{aligned}$$

(7.8)

In the final expression here, we have only explicitly written those terms that come from expanding the terms in the exponent in Eq. (7.7) up to cubic order. We have ordered the terms in Eq. (7.8) by increasing powers of $t$, and then by decreasing powers of $\sigma$, although shortly we will let $X$, $Y$ and $t$ themselves all be dependent on $\sigma$, and so the relative sizes of the terms will change. Note that we chose to factor out two terms (the difference between $g$ and $h$) for later convenience.

Let us now introduce some new coordinates $x$ and $y$, where again we scale the real and imaginary parts in different ways:

$$\begin{aligned} X&= a(\sigma )x + c(\sigma ), \nonumber \\ Y&= b(\sigma )y, \end{aligned}$$

(7.9)

in which the scalings and shift are given by (see [10])

$$\begin{aligned} a(\sigma )&\equiv \frac{\sigma }{\sqrt{6\log \sigma }}, \nonumber \\ b(\sigma )&\equiv \frac{\sigma ^{3/2}}{(6\log \sigma )^{1/4}}, \nonumber \\ c(\sigma )&\equiv a(\sigma )\Big ( 3 \log \sigma - \tfrac{5}{4} \log (6\log \sigma ) - \log (2\pi ) \Big ). \end{aligned}$$

(7.10)

We can choose how to scale $t$, since this will be an integration variable in the expressions for the limiting kernels. In order to make the analysis as straightforward as possible, it turns out that the following, simple scaling is optimal:

$$\begin{aligned} t = d(\sigma ) u, \end{aligned}$$

(7.11)

where

$$\begin{aligned} d(\sigma ) \equiv \sigma . \end{aligned}$$

(7.12)

So now consider the function

$$\begin{aligned} m(x,y,u,\sigma )&\equiv h(X + iY, t, \sigma ) \nonumber \\&= h\Big (a(\sigma )x + c(\sigma ) + ib(\sigma )y, d(\sigma )u, \sigma \Big ). \end{aligned}$$

(7.13)

On substituting into Eq. (7.8), we get a very large number of terms in the exponent (almost 80, in fact). However, most of these will vanish in the limit $\sigma \rightarrow \infty$, so we will retain only those that do not. After some simplification, we arrive at our second scaling limit of the deformed Airy function,

$$\begin{aligned} m(x,y,u,\sigma )&\sim \frac{(6 \log \sigma )^{5/8}}{\sigma ^{7/4}} \exp \left[ {} - \frac{x+y^2}{2} - \frac{u^2}{2} - \frac{u\sqrt{6\log \sigma }}{2} - \frac{iuy\sigma ^{3/2}}{(6\log \sigma )^{1/4}} \right] . \end{aligned}$$

(7.14)

For the third scaling limit, we use the asymptotic behaviour of the Airy function Eq. 10.4.60 of [47]

$$\begin{aligned} {{\mathrm{Ai}}}(-w) \sim \frac{\sin \left( \tfrac{2}{3} w^{3/2} + \frac{\pi }{4} \right) }{\sqrt{\pi } \, w^{1/4}} \end{aligned}$$

(7.15)

for large positive $w$ to show that, for $\alpha >0$ and at large real $w$, (with $z$, $\sigma$ and $\alpha$ fixed)

$$\begin{aligned} {{\mathrm{Aid}}}\left( \frac{z}{w} - \alpha w^2, \frac{\sigma }{w} \right)&\sim \frac{\exp [-\alpha \sigma ^2/2]}{\sqrt{\pi } \, \alpha ^{1/4}\sqrt{w}} \, \sin \left( \tfrac{2}{3} \alpha ^{3/2} w^3 \left( 1 - \frac{z}{\alpha w^3} - \frac{\sigma ^4}{4\alpha w^6} \right) ^{3/2} + \frac{\pi }{4} \right) \nonumber \\&\sim \frac{\exp [-\alpha \sigma ^2/2]}{\sqrt{\pi } \, \alpha ^{1/4}\sqrt{w}} \, \sin \left( \tfrac{2}{3} \alpha ^{3/2} w^3 - \sqrt{\alpha } \, z + \frac{\pi }{4} \right) . \end{aligned}$$

(7.16)

Appendix: 2 Hermitian Limit of the $\beta =4$ Matrix-kernel

As a check in this appendix we will take the Hermitian limit ${\sigma \rightarrow 0}$ of the $\beta =4$ matrix-kernel elements Eq. (4.21) as they appear in Eq. (4.4). Since all the elements of the matrix-kernel coincide to leading order, this leads to the appearance of first and second order derivatives acting on the Hermitian limit of the kernel. The same problem was encountered previously when taking the Hermitian limit of the $\beta =4$ kernel [48] in the microscopic hard-edge scaling limit, starting from the (interpolating) Bessel kernel in the complex plane [49]. Because we will follow [48] closely, we can be brief. There it was shown by using properties of the Pfaffian that the matrix-kernel inside Eq. (4.4) can be changed as follows, without changing the value of the Pfaffian:

$$\begin{aligned}&\left( \begin{array}{ll} \mathcal {K}(z_i,z_j) &{} \mathcal {K}(z_i,z_j^*) \\ \mathcal {K}(z_i^*,z_j) &{} \mathcal {K}(z_i^*,z_j^*) \end{array} \right) \rightarrow \nonumber \\&\quad \left( \begin{array}{ll} \mathcal {K}(z_i,z_j) - \mathcal {K}(z_i,z_j^*) - \mathcal {K}(z_i^*,z_j) + \mathcal {K}(z_i^*, z_j^*)) &{} \frac{1}{2}( \mathcal {K}(z_i,z_j)- \mathcal {K}(z_i^*,z_j) + \mathcal {K}(z_i,z_j^*) - \mathcal {K}(z_i^*, z_j^*) ) \\ \frac{1}{2}( \mathcal {K}(z_i,z_j) - \mathcal {K}(z_i,z_j^*) + \mathcal {K}(z_i^*,z_j) - \mathcal {K}(z_i^*, z_j^*)) &{} \frac{1}{4}( \mathcal {K}(z_i,z_j) + \mathcal {K}(z_i,z_j^*) + \mathcal {K}(z_i^*,z_j) + \mathcal {K}(z_i^*, z_j^*)) \end{array} \right) \!.\nonumber \\ \end{aligned}$$

(8.1)

We have suppressed all labels here, partly for simplicity, and partly because the replacement is an identity both before and after taking the large-$N$ limit. We will now expand the matrix-kernel elements from Eq. (4.21) as in Sect. 4.1, but this time to linear order in both $Y_1$ and $Y_2$, with $Y_{1,2},\sigma \ll 1$.

$$\begin{aligned}&\mathcal {K}_{\text {Ai}}^{(4)}(Z_1,Z_2^*) \approx \frac{-i\sqrt{|Y_1Y_2|}}{4\sqrt{\pi }\,\sigma ^3} \, \exp \left[ - \, \frac{Y_1^2+Y_2^2}{2\sigma ^2} \right] \nonumber \\&\qquad \times \left\{ \int _0^{\infty } ds \int _0^s dt \, \Big ( {{\mathrm{Ai}}}(X_2+s)-iY_2{{\mathrm{Ai}}}'(X_2+s) \Big ) \Big ( {{\mathrm{Ai}}}(X_1+t)+iY_1{{\mathrm{Ai}}}'(X_1+t) \Big ) \right. \nonumber \\&\qquad \left. {} - \Big ( {{\mathrm{Ai}}}(X_1+s)+iY_1{{\mathrm{Ai}}}'(X_1+s) \Big ) \Big ( {{\mathrm{Ai}}}(X_2+t)-iY_2{{\mathrm{Ai}}}'(X_2+t) \Big ) \right\} \nonumber \\&\quad \equiv v(Y_1,Y_2)\, \Big [ T_1(X_1,X_2) + iY_1T_2(X_1,X_2) + iY_2T_3(X_1,X_2) + Y_1Y_2T_4(X_1,X_2) \Big ], \end{aligned}$$

(8.2)

where

$$\begin{aligned} v(Y_1,Y_2)&=\frac{-i\sqrt{|Y_1Y_2|}}{4\sqrt{\pi }\,\sigma ^3} \, \exp \left[ - \, \frac{Y_1^2+Y_2^2}{2\sigma ^2} \right] ,\nonumber \\ T_1(X_1,X_2)&= \int _0^{\infty } ds \int _0^s dt \, \Big [ {{\mathrm{Ai}}}(X_2+s){{\mathrm{Ai}}}(X_1+t) - {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}(X_2+t) \Big ], \nonumber \\ T_2(X_1,X_2)&= \int _0^{\infty } ds \int _0^s dt \, \Big [ {{\mathrm{Ai}}}(X_2+s){{\mathrm{Ai}}}'(X_1+t) - {{\mathrm{Ai}}}'(X_1+s){{\mathrm{Ai}}}(X_2+t) \Big ] \nonumber \\&= 2\int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}(X_2+s) -{{\mathrm{Ai}}}(X_1) \int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_2+s), \nonumber \\ T_3(X_1,X_2)&= \int _0^{\infty } ds \int _0^s dt \, \Big [ -{{\mathrm{Ai}}}'(X_2+s){{\mathrm{Ai}}}(X_1+t) + {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}'(X_2+t) \Big ] \nonumber \\&= 2\int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}(X_2+s) -{{\mathrm{Ai}}}(X_2) \int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_1+s),\nonumber \\ T_4(X_1,X_2)&=\int _0^{\infty } ds \int _0^s dt \, \Big [ {{\mathrm{Ai}}}'(X_2+s){{\mathrm{Ai}}}'(X_1+t) - {{\mathrm{Ai}}}'(X_1+s){{\mathrm{Ai}}}'(X_2+t) \Big ] \nonumber \\&= \int _0^{\infty } ds \, \Big [ {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}'(X_2+s) - {{\mathrm{Ai}}}(X_2+s){{\mathrm{Ai}}}'(X_1+s) \Big ]. \end{aligned}$$

(8.3)

Whilst the integration in $T_4$ is straightforward, in $T_2$ and $T_3$ we have again used Eq. (4.25). Applying the shift Eq. (8.1) under the Pfaffian we obtain

$$\begin{aligned}&\lim _{\sigma ,Y\ll 1}R_{k,\text {Ai}}^{(4)}(Z_1,\ldots ,Z_k) \approx {{\mathrm{Pf}}}_{i,j=1,\ldots ,k} \left[ \!\left( \! \begin{array}{ll} -4Y_iY_jv(Y_i,Y_j)T_4(X_i,X_j) &{}2iY_iv(Y_i,Y_j)T_2(X_i,X_j)\\ -2iY_jv(Y_i,Y_j)T_3(X_i,X_j) &{}v(Y_i,Y_j)T_1(X_i,X_j)\\ \end{array} \!\right) \!\right] \nonumber \\&\quad =\prod _{j=1}^k\frac{2Y_j^2}{\sqrt{\pi }\,\sigma ^3} \, \exp \left[ - \, \frac{Y_j^2}{\sigma ^2} \right] {{\mathrm{Pf}}}_{i,j=1,\ldots ,k} \left[ \left( \begin{array}{ll} -\tfrac{1}{4} T_4(X_i,X_j) &{}\tfrac{1}{4} T_2(X_i,X_j)\\ -\tfrac{1}{4} T_3(X_i,X_j) &{}-\tfrac{1}{4} T_1(X_i,X_j)\\ \end{array} \right) \right] . \end{aligned}$$

(8.4)

In the second step we have taken out all the $Y$-dependent factors, which, using Eq. (4.27), will give rise to the product $\displaystyle \prod _{j=1}^k\delta (Y_j)$ in the limit $\sigma \rightarrow 0$ as expected. To see the matching with the known result for $\beta =4$ with real eigenvalues we note that

$$\begin{aligned} T_2(X_1,X_2)=T_3(X_2,X_1)=2\int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}(X_2+s) -{{\mathrm{Ai}}}(X_1) \int _{X_2}^{\infty } ds \, {{\mathrm{Ai}}}(s),\nonumber \\ \end{aligned}$$

(8.5)

where the first part is proportional to the $\beta =2$ Airy kernel of real eigenvalues, Eq. (3.29). Furthermore we can show that

$$\begin{aligned} T_4(X_1,X_2) = 2\int _0^{\infty } ds \, {{\mathrm{Ai}}}(X_1+s){{\mathrm{Ai}}}'(X_2+s) + {{\mathrm{Ai}}}(X_2){{\mathrm{Ai}}}(X_1) =\frac{\partial }{\partial X_2}T_2(X_1,X_2),\nonumber \\ \end{aligned}$$

(8.6)

and

$$\begin{aligned} T_1(X_1,X_2) =2 \int _{X_2}^{X_1}dt \int _0^{\infty } ds \, {{\mathrm{Ai}}}(t+s){{\mathrm{Ai}}}(X_2+s)- \int _{X_2}^{X_1}dt{{\mathrm{Ai}}}(t) \int _{X_2}^{\infty } ds \, {{\mathrm{Ai}}}(s), \end{aligned}$$

(8.7)

the latter of which can be verified by drawing a sketch of the $st$-plane showing the regions where there are contributions to the integrals, and comparing with the original form of $T_1$ in Eq. (8.3). Our final result for the limiting matrix kernel thus agrees with the literature, see e.g. page 162 of [42].

Appendix: 3 Some Properties of an Elementary Integral

Let us define the function for $x \in \mathbb {R}$, $\tau > 0$ and integer $j \ge 0$

$$\begin{aligned} I_j(x;\tau ) \equiv \int _{-\infty }^{\infty } dt \, {{\mathrm{sgn}}}(x-t) w^{(1)}(t) H_j \left( \frac{t}{\sqrt{2\tau }} \right) , \end{aligned}$$

(9.1)

where, for real $x$,

$$\begin{aligned} w^{(1)}(x) \equiv w^{(1)}(x;\tau ) = \exp \left[ - \, \frac{x^2}{2(1+\tau )} \right] , \end{aligned}$$

(9.2)

and $H_j(x)$ is the Hermite polynomial of degree $j$ orthogonal with respect to $\exp [-x^2]$. To determine a useful recurrence relation involving the $I_j(x;\tau )$, we begin by integrating Eq. (9.1) by parts using Eq. 8.952.1 of [45]:

$$\begin{aligned}&I_j(x;\tau ) = \left[ {{\mathrm{sgn}}}(x-t) w^{(1)}(t) \frac{\sqrt{2\tau }}{2(j+1)} H_{j+1} \left( \frac{t}{\sqrt{2\tau }} \right) \right] _{-\infty }^{\infty } \nonumber \\&\qquad {} - \frac{\sqrt{2\tau }}{2(j+1)} \int _{-\infty }^{\infty } dt \, H_{j+1} \left( \frac{t}{\sqrt{2\tau }} \right) w^{(1)}(t) \left\{ -2\delta (x-t) + {{\mathrm{sgn}}}(x-t) \left( \frac{-t}{1+\tau } \right) \right\} \nonumber \\&\quad = \frac{\sqrt{2\tau }}{j+1} H_{j+1} \left( \frac{x}{\sqrt{2\tau }} \right) w^{(1)}(x) \nonumber \\&\qquad + \frac{\sqrt{\tau }}{\sqrt{2}(1+\tau )(j+1)} \int _{-\infty }^{\infty } dt \, t \, {{\mathrm{sgn}}}(x-t) w^{(1)}(t) H_{j+1} \left( \frac{t}{\sqrt{2\tau }} \right) . \end{aligned}$$

(9.3)

From Eq. 8.952.2 of [45] we have the recurrence relation for Hermite polynomials

$$\begin{aligned} t \, H_n\left( \frac{t}{\sqrt{2\tau }} \right) = \sqrt{2\tau } \left\{ \tfrac{1}{2}H_{n+1}\left( \frac{t}{\sqrt{2\tau }} \right) + n \, H_{n-1}\left( \frac{t}{\sqrt{2\tau }} \right) \right\} , \end{aligned}$$

(9.4)

and hence

$$\begin{aligned} I_j(x;\tau ) \!=\! \frac{\sqrt{2\tau }}{j+1} H_{j+1} \left( \frac{x}{\sqrt{2\tau }} \right) w^{(1)}(x) \!+\! \frac{\tau }{(1\!+\!\tau )(j\!+\!1)} \left\{ \tfrac{1}{2} I_{j+2}(x;\tau ) \!+\! (j+1)I_j(x;\tau ) \right\} \ , \end{aligned}$$

(9.5)

which can be rearranged to give the following recurrence relation for the $I_j(x;\tau )$ themselves:

$$\begin{aligned} I_j(x;\tau )&= \frac{\sqrt{2\tau }(1+\tau )}{j+1} H_{j+1} \left( \frac{x}{\sqrt{2\tau }} \right) w^{(1)}(x) + \frac{\tau }{2(j+1)} \, I_{j+2}(x;\tau ). \end{aligned}$$

(9.6)

We can evaluate the integral $I_j(x;\tau )$ by repeated application of Eq. (9.6), to give for odd $j$:

$$\begin{aligned} I_j(x;\tau ) = - 2(1+\tau ) \left( \frac{2}{\tau } \right) ^{j/2} (j-1)!! \, w^{(1)}(x) \sum _{k=0}^{(j-1)/2} \frac{1}{(2k)!!} \left( \frac{\tau }{2} \right) ^k H_{2k}\left( \frac{x}{\sqrt{2\tau }}\right) ,\qquad \end{aligned}$$

(9.7)

and for even $j$:

$$\begin{aligned} I_j(x;\tau )&= \left( \frac{2}{\tau } \right) ^{j/2} (j-1)!!\nonumber \\&\times \left\{ I_0(x;\tau ) - 2(1+\tau )w^{(1)}(x) \sum _{k=0}^{j/2-1} \frac{1}{(2k+1)!!} \left( \frac{\tau }{2} \right) ^{k+1/2} H_{2k+1}\left( \frac{x}{\sqrt{2\tau }}\right) \right\} ,\nonumber \\ \end{aligned}$$

(9.8)

where $I_0(x;\tau )$ can be written in terms of the error function as

$$\begin{aligned} I_0(x;\tau ) = \sqrt{2\pi (1+\tau )} \, {{\mathrm{erf}}}\left( \frac{x}{\sqrt{2(1+\tau )}} \right) . \end{aligned}$$

(9.9)

Appendix: 4 Hermitian Limit of the $\beta =1$ Matrix-Kernel

As with the $\beta =4$ case in Appendix 2, we verify that the Hermitian limit $\sigma \rightarrow 0$ of the matrix-kernel (and hence of all $k$-point correlation functions) agrees with known results in the literature. Since we do not have the discontinuity associated with the $\beta =4$ ensemble, here we can, in all cases, let $\sigma \rightarrow 0$ at fixed arguments (eigenvalues) $Z_j=X_j+iY_j$. Indeed, note that no new Dirac delta functions appear in any of the following limits, since these are already present in the non-Hermitian case, see Eqs. (5.34) and (5.35).

It is useful to introduce the following general result for the pointwise limit of the complementary error function:

$$\begin{aligned} \lim _{\sigma \rightarrow 0} {{\mathrm{erfc}}}\left( \frac{|Y|}{\sigma } \right) = \delta _{Y0}, \end{aligned}$$

(10.1)

where, for $x,y \in \mathbb {R}$, we define the Kronecker delta function as

$$\begin{aligned} \delta _{xy} = {\left\{ \begin{array}{ll} \,\, 1 &{} \text {if }x = y, \\ \,\, 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

(10.2)

• (i) We begin with $\hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2)$, given in Eq. (5.38). On expanding the $e^{\sigma ^2 t}$ inside the integral, and then letting $\sigma \rightarrow 0$, this gives

  $$\begin{aligned}&\lim _{\sigma \rightarrow 0} \hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2)\nonumber \\&\quad = \frac{(X_1-X_2)}{4} \, \int _0^{\infty } dt \, t \, {{\mathrm{Ai}}}(X_1+t) {{\mathrm{Ai}}}(X_2+t) \, \delta _{Y_1 0} \delta _{Y_2 0} \nonumber \\&\quad = \frac{1}{4} \left\{ \int _0^{\infty } dt \, t \, \big [ {{\mathrm{Ai}}}''(X_1+t) - t\,{{\mathrm{Ai}}}(X_1+t) \big ] \, {{\mathrm{Ai}}}(X_2+t) - (X_1 \leftrightarrow X_2) \right\} \, \delta _{Y_1 0} \delta _{Y_2 0}\nonumber \\&\quad = \frac{1}{4} \int _0^{\infty } dt \, t \, \big [ {{\mathrm{Ai}}}''(X_1+t){{\mathrm{Ai}}}(X_2+t) - (X_1 \leftrightarrow X_2) \big ] \, \delta _{Y_1 0} \delta _{Y_2 0}\nonumber \\&\quad = \frac{1}{4} \left\{ - \int _0^{\infty } dt \, {{\mathrm{Ai}}}'(X_1+t) \big [ t\,{{\mathrm{Ai}}}'(X_2+t) + {{\mathrm{Ai}}}(X_2+t) \big ] - (X_1 \leftrightarrow X_2) \right\} \, \delta _{Y_1 0} \delta _{Y_2 0}\nonumber \\&\quad = \frac{1}{4} \left\{ - \int _0^{\infty } dt \, {{\mathrm{Ai}}}'(X_1+t) {{\mathrm{Ai}}}(X_2+t) - (X_1 \leftrightarrow X_2) \right\} \, \delta _{Y_1 0} \delta _{Y_2 0}\nonumber \\&\quad = \frac{1}{2} \left\{ \frac{\partial }{\partial X_2} \mathcal {K}_{\text {Ai,Herm}}^{(2)}(X_1,X_2) + \tfrac{1}{2} \, {{\mathrm{Ai}}}(X_1){{\mathrm{Ai}}}(X_2) \right\} \, \delta _{Y_1 0} \delta _{Y_2 0}, \end{aligned}$$

  (10.3)

  where we used the fact that ${{\mathrm{Ai}}}''(X_1+t)=(X_1+t){{\mathrm{Ai}}}(X_1+t)$ in the first step, cancelled terms, integrated by parts, cancelled more terms, and then integrated only the $X_1$-term by parts. We also used the result Eq. (10.1) that the limit of the complementary error function involves the delta function defined in Eq. (10.2). In the final line, $\mathcal {K}_{\text {Ai,Herm}}^{(2)}(X_1,X_2)=\int _0^{\infty } dt \, {{\mathrm{Ai}}}(X_1+t) {{\mathrm{Ai}}}(X_2+t)$ is the Hermitian limit of the $\beta =2$ kernel, see Eq. (3.30).

• (ii) For $G_{\text {Ai}}^{\mathbb {R}\,(1)}(Z_1,X_2)$, we take $\sigma \rightarrow 0$ in Eq. (5.49). This gives

  $$\begin{aligned} - \lim _{\sigma \rightarrow \infty } G_{\text {Ai}}^{\mathbb {R}\,(1)}(Z_1,X_2) = \left\{ \mathcal {K}_{\text {Ai,Herm}}^{(2)}(X_1,X_2) + \tfrac{1}{2} \, {{\mathrm{Ai}}}(X_1)\left( 1 - \int _0^{\infty } dt \, {{\mathrm{Ai}}}(X_2 + t) \right) \right\} \, \delta _{Y_1 0}.\nonumber \\ \end{aligned}$$

  (10.4)

• (iii) For $G_{\text {Ai}}^{\mathbb {C}\,(1)}(Z_1,Z_2)$, we have

  $$\begin{aligned} G_{\text {Ai}}^{\mathbb {C}\,(1)}(Z_1,Z_2) = 2i \, {{\mathrm{sgn}}}(Y_2) \hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2^*). \end{aligned}$$

  (10.5)

  The pre-kernel $\hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2^*)$ only has a non-zero limit when the arguments are both real (i.e. when $Y_1=Y_2=0$). But in this case, ${{\mathrm{sgn}}}(Y_2)=0$, and so $G_{\text {Ai}}^{\mathbb {C}\,(1)}(Z_1,Z_2) \rightarrow 0$ for all arguments.

• (iv) For $W_{\text {Ai}}^{\mathbb {R}\mathbb {R}\,(1)}(X_1,X_2)$, we simply set $\sigma = 0$ in Eq. (5.62):

  $$\begin{aligned}&- \lim _{\sigma \rightarrow 0} W_{\text {Ai}}^{\mathbb {R}\mathbb {R}\,(1)}(X_1,X_2)\nonumber \\&\quad = \lim _{\sigma \rightarrow 0} \Big \{ - 2 A(X_1,X_2) + B(X_1)B(X_2) + B(X_2) - B(X_1) \Big \} \nonumber \\&\quad = 2 \left\{ \int _{X_2}^{X_1}\! dt \, \mathcal {K}_{\text {Ai,Herm}}^{(2)}(t,X_2) + \tfrac{1}{2} \left( \int _{X_2}^{X_1}\! dt \, {{\mathrm{Ai}}}(t) \right) \! \left( 1 - \int _{X_2}^{\infty }\! dt \, {{\mathrm{Ai}}}(t) \right) \! \right\} . \end{aligned}$$

  (10.6)

  The second step here can be proved with a few lines of easy manipulation.

• (v) For $W_{\text {Ai}}^{\mathbb {R}\mathbb {C}\,(1)}(X_1,Z_2)$, we have

  $$\begin{aligned} W_{\text {Ai}}^{\mathbb {R}\mathbb {C}\,(1)}(X_1,Z_2) = 2i \, {{\mathrm{sgn}}}(Y_2) G_{\text {Ai}}^{\mathbb {R}\,(1)}(Z_2^*,X_1). \end{aligned}$$

  (10.7)

  The function $G_{\text {Ai}}^{\mathbb {R}\,(1)}(Z_2^*,X_1)$ only has a non-zero limit when the first argument is real (i.e. when $Y_2=0$). But in this case, ${{\mathrm{sgn}}}(Y_2)=0$, and so $W_{\text {Ai}}^{\mathbb {R}\mathbb {C}\,(1)}(X_1,Z_2) \rightarrow 0$ for all arguments.

• (vi) For $W_{\text {Ai}}^{\mathbb {C}\mathbb {C}\,(1)}(Z_1,Z_2)$, we have

  $$\begin{aligned} W_{\text {Ai}}^{\mathbb {C}\mathbb {C}\,(1)}(Z_1,Z_2) = -2i \, {{\mathrm{sgn}}}(Y_1) G_{\text {Ai}}^{\mathbb {C}\,(1)}(Z_1^*,Z_2). \end{aligned}$$

  (10.8)

  But $G_{\text {Ai}}^{\mathbb {C}\,(1)}(Z_1^*,Z_2)\rightarrow 0$, and so $W_{\text {Ai}}^{\mathbb {C}\mathbb {C}\,(1)}(Z_1,Z_2)\rightarrow 0$ for all arguments.

• (vii) The Hermitian limit of the bivariate weight function $\mathcal {F}_{\infty }^{(1)}(Z_1,Z_2)$ is trivial, since $\mathcal {F}_{\infty }^{(1)}(Z_1,Z_2)$ does not depend on $\sigma$. We have simply

  $$\begin{aligned} \lim _{\sigma \rightarrow 0} \mathcal {F}_{\infty }^{(1)}(Z_1,Z_2) = 2i\delta ^{(2)}(Z_1-Z_2^*){{\mathrm{sgn}}}(Y_1) + \delta (Y_1)\delta (Y_2){{\mathrm{sgn}}}(X_2-X_1).\qquad \end{aligned}$$

  (10.9)

In fact, we can drop the first term completely in the Hermitian limit, since it always gets multiplied by $\hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2;\sigma =0) \propto \delta _{Y_1 0}$ when evaluating the Pfaffian to determine the correlation functions.

We expect the $\sigma \rightarrow 0$ limits to be consistent with those in the literature, e.g. on page 162 of [42]; our limit of $G_{\text {Ai}}^{(1)}(Z_1,Z_2)$ matches [42] precisely. Our limit of $\hat{\mathcal {K}}_{\text {Ai}}^{(1)}(Z_1,Z_2)$ is one half of that given in [42], but our limit of $W_{\text {Ai}}^{(1)}(Z_1,Z_2)$ is twice what is given in [42]. It therefore follows that, once we evaluate the Pfaffian, we will get precisely the same $k$-point correlation functions for all values of $k$.