Recursion scheme for the largest $\beta$-Wishart-Laguerre eigenvalue and Landauer conductance in quantum transport

TL;DR

This paper introduces a recursion scheme for computing the largest eigenvalue distribution of the Wishart-Laguerre ensemble, with applications in quantum transport and communication, providing exact results where previous methods were limited.

Contribution

It presents a novel recursion method for explicit eigenvalue distribution calculation, applicable to both original and fixed-trace models, improving over symbolic determinant-based approaches.

Findings

Effective recursion scheme for eigenvalue distribution

Exact Landauer conductance distribution solution

Applications in quantum communication and entanglement

Abstract

The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed by Dyson parameter $\beta$ and Laguerre parameter $a$, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalization of the Selberg integral, we provide an effective recursion scheme to compute this distribution explicitly in both the original model, and a fixed-trace variant, for $a,\beta$ non-negative integers and finite matrix size. For $\beta = 2$ this circumvents known symbolic evaluation based on determinants which become impractical for large dimensions. Our exact results have immediate applications in the areas of multiple channel communication and bipartite entanglement. Moreover, we are also led to the exact solution of a long standing problem of finding a general result for Landauer conductance distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far, exact closed-form results for this were available only in the Fourier-Laplace space or could be obtained on a case-by-case basis.

Equations

$$\mathcal{P}(x_1,\dots ,x_n)=\frac{1}{W}\prod _{l=1}^n{x}_l^a{e}^{-\beta x_l/2}\prod _{1\le j<k\le n}|x_k-x_j{|}^{\beta },$$

$$W={\left(\frac{2}{\beta }\right)}^{\gamma }\prod _{j=0}^{n-1}\frac{\Gamma \left(\frac{\beta (j+1)}{2}+1\right)\Gamma \left(\frac{\beta j}{2}+a+1\right)}{\Gamma \left(\frac{\beta }{2}+1\right)},$$

$$P(x)=n{\int }_0^{x}d{x}_2\cdots {\int }_0^{x}d{x}_n\mathcal{P}(x,x_2,\dots ,x_n),$$

$$Q(x)={\int }_0^{x}d{x}_1\cdots {\int }_0^{x}d{x}_n\mathcal{P}(x_1,...,x_n)={\int }_0^{x}d{x}^{\prime }P(x^{\prime }).$$

$$P(x)=\sum _{j=1}^n{e}^{-j\beta x/2}\sum _{k=a}^{ja+j(n-j)\beta }{c}_{jk}{x}^k,$$

$$Q(x)=\sum _{j=0}^n{e}^{-j\beta x/2}\sum _{k=0}^{ja+j(n-j)\beta }{d}_{jk}{x}^k.$$

$${\mathcal{P}}_F(y_1,\dots ,y_n)=\frac{1}{W_F}\delta \left(\sum _i{y}_i-1\right)\prod _{l=1}^n{y}_l^a\prod _{1\le j<k\le n}|y_k-y_j{|}^{\beta },$$

$$P_F(y)=\frac{2}{\beta }\Gamma (\gamma )\sum _{j=1}^n\Theta (1-jy)\sum _{k=a}^{ja+j(n-j)\beta }\frac{c_{jk}}{\Gamma (\gamma -k-1)}{\left(\frac{2y}{\beta }\right)}^k(1-jy{)}^{\gamma -k-2},$$

$$Q_F(y)=\Gamma (\gamma )\sum _{j=0}^n\Theta (1-jy)\sum _{k=0}^{ja+j(n-j)\beta }\frac{d_{jk}}{\Gamma (\gamma -k)}{\left(\frac{2y}{\beta }\right)}^k(1-jy{)}^{\gamma -k-1},$$

$$\prod _{1\le j<k\le n}|x_k-x_j{|}^{\beta }=\sum _{\kappa =({\kappa }_1,\dots ,{\kappa }_n)}{\alpha }_{\kappa }\prod _{l=1}^n{x}_l^{{\kappa }_l},$$

$$Q(x)=\frac{1}{W}\sum _{\kappa }{\alpha }_{\kappa }\prod _{l=1}^n\left({\int }_0^{x}d{x}_l{x}_l^{a+{\kappa }_l}{e}^{-\beta x_l/2}\right).$$

$${\int }_0^{x}ds{s}^a{e}^{-\beta s/2}=a!{\left(\frac{2}{\beta }\right)}^{a+1}\left[1-e^{-\beta x/2}\sum _{k=0}^a\frac{1}{k!}{\left(\frac{\beta x}{2}\right)}^k\right].$$

$$L_{p,\nu }^{(\alpha )}(x)=\frac{p!(\nu -p)!}{\nu !}{\int }_0^{x}d{t}_1\cdots {\int }_0^{x}d{t}_{\nu }\prod _{l=1}^{\nu }{t}_l^{{\lambda }_1}{e}^{-\lambda t_l}|x-t_l{|}^{\alpha }$$

$$\times \prod _{1\le j<k\le \nu }|t_k-t_j{|}^{2\lambda }{e}_p(x-t_1,\dots ,x-t_{\nu }),$$

$$\lambda (\nu -p)L_{p+1,\nu }^{(\alpha )}(x)=[\lambda (\nu -p)x+B_p]L_{p,\nu }^{(\alpha )}(x)$$

$$+x\frac{d}{dx}{L}_{p,\nu }^{(\alpha )}(x)-D_{p}x{L}_{p-1,\nu }^{(\alpha )}(x),$$

$$B_p$$

$${\int }_0^{x}d{x}_1{x}_1^a{e}^{-\beta x_1/2}|x-x_1{|}^{\beta }.$$

$${\int }_0^{x}dx{x}_2^a{e}^{-\beta x_2/2}{\int }_0^{x_2}d{x}_1{x}_1^a{e}^{-\beta x_1/2}|x_2-x_1{|}^{\beta },$$

$${\int }_0^{x}d{x}_2{x}_2^a{e}^{-\beta x_2/2}{\int }_0^{x}d{x}_1{x}_1^a{e}^{-\beta x_1/2}\prod _{1\le j<k\le 3}|x_k-x_j{|}^{\beta }$$

$$g=\sum _{j=1}^n{T}_j,$$

$$P(T_1,....,T_n)\propto \prod _{l=1}^n{T}_l^a\prod _{1\le j<k\le n}|T_k-T_j{|}^{\beta },$$

$$P_g(g)={\int }_0^{1}d{T}_1\cdots {\int }_0^{1}d{T}_n\delta \left(g-\sum _{j=1}^n{T}_j\right)P(T_1,\dots ,T_n).$$

$$P_g(g)=\frac{1}{\Gamma (\gamma )}K{g}^{\gamma -1}{Q}_F(1/g);$$

$$K=\prod_{l=0}^{n-1}\frac{\Gamma\Big{(}a+\frac{\beta(n+l-1)}{2}+2\Big{)}}{\Gamma(\frac{\beta l}{2}+1)}.$$

$$P_g(g)=K\sum _{j=0}^n\Theta (g-j)\sum _{k=0}^{ja+j(n-j)\beta }\frac{d_{jk}}{\Gamma (\gamma -k)}{\left(\frac{2}{\beta }\right)}^k(g-j{)}^{\gamma -k-1}.$$

$${\rho }_{(j)}(x_1,\dots ,x_j)=\frac{n!}{(n-j)!}{\int }_0^{\infty }d{x}_{j+1}\cdots {\int }_0^{\infty }d{x}_n\mathcal{P}(x_1,...,x_n),$$

$$Q(x)=1+\sum _{j=1}^n\frac{(-1{)}^j}{j!}{\int }_x^{\infty }d{x}_1\cdots {\int }_x^{\infty }d{x}_j{\rho }_{(j)}(x_1,\dots ,x_j).$$

$$\prod _{l=1}^j{x}_l^{a+\beta (n-j)}{e}^{-\beta x_l/2}\prod _{1\le p<q\le j}|x_p-x_q{|}^{\beta }.$$

Full text

1 Introduction

The Wishart-Laguerre ensemble constitutes an important class of random matrices and has been of immense usefulness in modeling a variety of problems in diverse topics [1, 2, 3]. The corresponding extreme eigenvalues, aside from being fundamental in multivariate statistics [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 15], play crucial roles in problems ranging from quantum entanglement in bipartite systems [16, 18, 17, 19], and electronic transport properties in mesoscopic systems [10, 20], to multichannel communication in wireless networks [21, 22, 23, 25, 26, 27, 24, 28, 29]. The asymptotic behaviour of these extreme eigenvalues and the corresponding gap probabilities have been explored extensively, leading to Tracy-Widom densities [30, 31, 32, 12, 13, 33] and large-deviation results [34, 35, 36, 37] of significant importance. In certain applications, however, one requires to go beyond universal results and seek exact and explicit computation of these distributions for finite matrix size. Typical theory in this connection is based on determinants or Pfaffians requiring symbolic computation [6, 10, 11, 14, 26, 15], and therefore becomes impractical to evaluate if large matrices are involved. An alternative approach for obtaining explicit expressions involves implementing certain recurrences. This has turned out to be very effective considering the availability of modern software packages which can handle recursive symbolic computation very efficiently. For the smallest eigenvalue of real Wishart-Laguerre matrices, in Refs. [38, 39] Edelman provided a recursion scheme which has subsequently been generalised to other ensembles and symmetry classes [10, 40, 1, 41, 42, 19, 43, 44]. However, extending this to the largest eigenvalue distribution has remained elusive due to its comparatively more convoluted mathematical structure [1].

In this work, we accomplish the task of providing an efficient recursion scheme for evaluating the probability density function (PDF) as well as the cumulative distribution function (CDF) of the largest eigenvalue of the integer $\beta$-Wishart-Laguerre ensemble for both unrestricted and fixed trace variants. These results apply at once to problems pertaining to multiple channel communication in wireless systems and bipartite entanglement in random pure states. Additionally, we exploit the connection of the largest eigenvalue of the fixed trace Wishart-Laguerre ensemble to the quantum conductance problem and solve a long-standing problem of obtaining a general closed-form result for the Landauer conductance distribution in a chaotic mesoscopic cavity with ideal leads. Thus far, exact closed-form results for this distribution were available only in the Fourier-Laplace space as a determinant or could be obtained on a case by case basis [47, 48, 46, 45, 49, 50, 52].

To present our findings, we begin by providing the general results for the functional form of the PDF and CDF of the Wishart-Laguerre largest eigenvalue, along with a brief discussion of their immediate applicability to the multiple channel communication and bipartite entanglement problems. We then briefly describe the machinery behind our proposed recursive approach and also present comparison of analytical and Monte Carlo simulation based results for a few examples. The application to the quantum conductance problem is discussed next, where we point out the implementation of recursion scheme to obtain exact Landauer conductance distribution. The analytical results in this case are contrasted with the numerical results obtained with the aid of scattering matrices modelled using circular ensemble of random matrices. Finally, we conclude with a brief summary of our work.

2 Exact distribution for the $\beta$-Wishart-Laguerre largest eigenvalue

The classical cases of Wishart-Laguerre ensemble include real, complex and quaternion positive-definite matrices of the form $GG^{\dagger}$, designated by the Dyson index $\beta=1,2$ and 4, respectively. In recent years, matrix models for continuous $\beta>0$ variants have been also worked out and have drawn considerable attention. In the general case, the joint PDF for Wishart-Laguerre eigenvalues ($0<x_{1},...,x_{n}<\infty$) is given by

[TABLE]

with $a>-1$. The partition function $W$ is known using the Selberg integral as [1, 2]

[TABLE]

where $\gamma=n[a+\beta(n-1)/2+1]$. The PDF and CDF of the largest eigenvalue are computed as [1]

[TABLE]

The CDF of the largest eigenvalue coincides with the gap probability $E_{n,\beta}(0;(x,\infty))$ of finding no eigenvalue in the domain $(x,\infty)$.

We demonstrate below that for positive integer $\beta$ and non-negative integer $a$, the sought PDF and CDF exhibit respective structures:

[TABLE]

[TABLE]

For complex matrices ($\beta=2$), the above structures have been pointed out in Ref. [22] in the context of multiple channel communication problem. However, the computation of the coefficients (which depend on $\beta,a$ and $n$ and evaluate to rational numbers) has remained a daunting task [22, 23, 24]. These earlier works have relied on evaluating the coefficients using the Hankel determinant of an $n$-dimensional matrix. This becomes prohibitive if $n$ becomes large (say, $\gtrsim 10$). As we will see below, our recursive approach facilitates the evaluation of PDF, CDF and coefficients in an efficient manner.

The fixed-trace variant of the Wishart-Laguerre ensemble is described by the joint density

[TABLE]

where the eigenvalues $(0\leq y_{1},...,y_{n}\leq 1)$ are constrained by the Dirac-delta condition, and $W_{F}=[1/\Gamma(\gamma)](\beta/2)^{\gamma}W$ is the corresponding partition function. The PDF and CDF of the largest eigenvalue for this ensemble follows from the Fourier-Laplace relationship with the unrestricted trace variant [1], equation (1), as

[TABLE]

[TABLE]

where $\Theta(z)$ denotes the Heaviside step function. It should be noted that the largest eigenvalue in this case also coincides with the scaled largest eigenvalue associated with (1), i.e., $y_{\mathrm{max}}=x_{\mathrm{max}}/(x_{1}+\dots+x_{n})$.

The above results apply directly to the multiple channel communication and bipartite entanglement problems. In multiple input multiple output (MIMO) communication, the channel matrix $H$ is the central object. It models the fading of the signal between $n_{t}$ and $n_{r}$ number of transmitting and receiving antennas, respectively. The singular values of the $n_{r}\times n_{t}$–dimensional $H$ matrix or, equivalently, the eigenvalues of $HH^{\dagger}$ (or $H^{\dagger}H)$ play a crucial role in deciding several quantities of interest which assess the performance of the MIMO system. In the cases of one-sided Gaussian and Rayleigh fadings, the eigenvalues are described by the density given in (1) with $\beta=1$ and 2, respectively, and the parameter $a=\beta(|n_{t}-n_{r}|+1)/2-1$. In Ref. [22, 23], it has been shown within a transmit-beamforcing model with maximum ratio combining (MRC) receivers that the maximum output signal-to-noise (SNR) is directly related to the largest eigenvalues of $HH^{\dagger}$. Consequently, the distribution of the largest eigenvalue has been used to obtain the statistics of symbol error probability (SEP), outage probability and ergodic channel capacity [22, 23]. Additionally, the distribution of the scaled largest eigenvalue finds applications in various hypothesis testing problems, both in statistics and in signal processing [53, 54, 55, 56, 57, 59, 58, 24].

In the problem of bipartite entanglement, one is interested in the properties of the reduced density matrix. To elaborate, consider Hilbert spaces $\mathcal{H}_{n},\mathcal{H}_{m}$ of dimensions $n$ and $m$ for the constituents with $n\leq m$. Moreover, let $|\phi\rangle$ be a pure state belonging to the composite system, i.e., $|\phi\rangle\in\mathcal{H}_{n}\otimes\mathcal{H}_{m}$. In case $|\phi\rangle$ is chosen randomly in a uniform manner, the Schmidt eigenvalues of the reduced density matrix $\rho=\tr_{m}(|\phi\rangle\langle\phi|)$ obtained by partial tracing over the part belonging to $\mathcal{H}_{m}$ are governed by the joint PDF (7) with $a=\beta(m-n+1)/2-1$ [1, 60]. The largest of these eigenvalues varies from $1/n$ to 1. In these two extremes, owing to the fixed trace restriction, all other eigenvalues assume a common value of $1/n$ and 0, respectively. These respective scenarios correspond to the two subsystems being maximally entangled and separable. The distribution of the largest eigenvalue of the fixed trace ensemble therefore carries important information regarding the entanglement between the subsystems of the composite bipartite system [18].

3 Recursion scheme

We now briefly describe the machinery behind the recursion scheme to obtain the CDF and PDF of the largest eigenvalue. To begin with, we will demonstrate that the multi-dimensional integral in (4) for $Q(x)$ does lead to the structure shown in (6). For even $\beta$, we expand $\prod_{j<k}|x_{k}-x_{j}|^{\beta}$ as a homogeneous multivariable polynomial of degree $\beta n(n-1)/2$, viz.,

[TABLE]

for some coefficients $\alpha_{\kappa}$. Here $\kappa$ is an integer partition involving up to $n$ parts and of fixed length such that $\sum_{i=1}^{n}\kappa_{i}=\beta n(n-1)/2$. With $\kappa_{1}\geq\kappa_{2}\geq\cdots$, it is known $\kappa_{j}\leq(n-j)\beta$. Substituting the above expansion in (4) yields

[TABLE]

Provided $a$ is a non-negative integer, the integral inside the above product can be evaluated as

[TABLE]

Therefore, eventually we wind up with the expansion of the form given in (6), but with the upper limit of the inner sum (over $k$) equal to $ja+j[n-(j+1)/2]\beta$. As shown in the Appendix A, this can be refined to $ja+j(n-j)\beta$ as in (6). Also, Eqs. (11) and (12) imply that for $\beta$ even $d_{j0}=(-1)^{j}\left({n\atop j}\right)$. For handling odd $\beta$, we order the eigenvalues so that $x_{1}>x_{2}>\dots x_{n}>0$, and then the expansion in (10) is valid for appropriate coefficients $\alpha_{\kappa}$. Moreover, the $Q(x)$ can be written as the iterated multi-dimensional integral $n!\int_{0}^{x}dx_{1}\cdots\int_{0}^{x_{n-2}}dx_{n-1}\int_{0}^{x_{n-1}}dx_{n}\mathcal{P}(x_{1},...,x_{n})$, leading again to (6). The expansion in (5) for the PDF $P(x)$ follows similarly by using Eqs. (10) and (12) in (3).

We next discuss a recursion scheme for the computation of the coefficients. We introduce the auxiliary function, generalising the Laguerre weighted Selberg integral [1, 2]

[TABLE]

where $e_{p}(t_{1},\ldots,t_{\nu})$ are the elementary symmetric polynomials. At the heart of the recursion lies a remarkable linear differential-difference equation [42, 43],

[TABLE]

where $p=0,1,...,\nu-1$ and

[TABLE]

Suppose $L_{0,\nu}^{(\alpha)}(x)$ is known. Application of the recurrence allows for the computation of $L_{\nu,\nu}^{(\alpha)}(x)$ which is identical to $L_{0,\nu}^{(\alpha+1)}(x)$. With $\nu=n$, $2\lambda=\beta$, iterating to the value $\alpha=\beta$ gives $Q(x)$.

The difficulty is that $L_{0,\nu}^{(\alpha)}(x)$ is known only for $\nu=1$. Fortunately, the relation between $P(x)$ and $Q(x)$ displayed in (4) permits an iteration also in $\nu$, as observed in a recursive computation of the largest eigenvalue PDF of the Gaussian orthogonal ensemble given by James [61]. Thus we begin with (12) to obtain $\int_{0}^{x}dx_{1}\,x_{1}^{a}e^{-\beta x_{1}/2}$. The recurrence in (3) can then be applied $\beta$ times to evaluate

[TABLE]

Next, we denote $x$ by $x_{2}$ in the above expression, multiply by $x_{2}^{a}e^{-\beta x_{2}/2}$, and integrate over $x_{2}$ from 0 to $x$ to obtain

[TABLE]

which is the same as half this double-integral with the upper limit of the inner integral changed to $x$ from $x_{2}$. Now we essentially repeat this procedure. Using the recurrence, knowledge of the above two dimensional integral is used to compute

[TABLE]

with $x_{3}=x$, which is then multiplied by $x_{3}^{a}e^{-\beta x_{3}/2}$, and integrated over $x_{3}$, and so on. Continuing, we eventually arrive at the expansion (6). An important point is that at all stages of the iteration the one-dimensional integral in (12) applies and the expressions can be written in an expansion analogous to (6). It should be also noted that the expansion for $P(x)$ appears in this procedure a step before the final integration is considered for $Q(x)$. Furthermore, when evaluating the expressions for a given $n$, the results for lower dimensions are also obtained in the process.

A Mathematica [62] code to implement the above described recursion scheme can be found in the supplementary material. It also extracts the coefficients $c_{jk}$ and $d_{jk}$ which can then be used for evaluating results for the fixed trace ensemble. As an example, we have considered $n=10,a=15$ and shown the plots for $\beta=1,2,3,4$ in Fig. 1. The analytical result based solid curves are contrasted against the numerical simulation based overlaid symbols and we can see an excellent agreement.

4 Exact distribution of Landauer conductance

One of the remarkable achievements of random matrix theory (RMT) has been in the field of quantum conductance in chaotic mesoscopic systems [63, 64, 65, 46, 47, 45, 48, 66]. Starting from its prediction of universal conductance fluctuation [63, 64, 65, 46, 45] to the modern day description of topological superconductors and Majorana fermions [66], RMT has had great success in modeling various charge transport related phenomena. Despite this, there remain several problems which have defied an exact solution. One such problem is working out the exact distribution of Landauer conductance in a chaotic mesoscopic cavity with ideal leads [47, 48, 46, 45, 49, 50, 51, 52]. Here we solve this problem by exploiting its mathematical connection with largest eigenvalue of the fixed trace Wishart-Laguerre ensemble.

The chaotic mesoscopic cavity is connected to an electron reservoir via two ideal leads. The electronic transport properties follow from the knowledge of the scattering matrix ($S$ matrix). In this case, the $S$ matrix can be modelled using Dyson’s circular ensemble [46, 47, 45, 67] (note that for nonideal leads the $S$-matrix has to be taken from a nonuniform measure given by the Poisson kernel [68, 45, 1, 69]). For given number of channels $n_{1}$ and $n_{2}$ in the two leads, the $S$ matrix is $n_{1}+n_{2}$ dimensional. The Landauer conductance, measured in units of $e^{2}/h$, is then given by $g=\mathrm{tr}(\Pi_{1}S\Pi_{2}S^{\dagger})$, where $\Pi_{1}$ is a projection matrix with elements $(\Pi_{1})_{j,k}=\delta_{jk}$ for $j\leq n_{1}$ and zero otherwise and $\Pi_{2}=\mathds{1}_{n_{1}+n_{2}}-\Pi_{1}$. For $\beta=4$, the $S$ matrix has quaternionic entries and $\Pi_{1},\Pi_{2}$ are accordingly modified. The Landauer conductance is equivalently described in terms of the transmission eigenvalues $(0\leq T_{1},...,T_{n}\leq 1)$ as [45, 70, 71]

[TABLE]

where $n=\min(n_{1},n_{2})$. We also define $m=\max(n_{1},n_{2})$. The joint PDF describing the transmission eigenvalues is given by [46, 47, 45, 67, 72]

[TABLE]

where $a=\beta(m-n+1)/2-1$. The above is a special case of the Jacobi ensemble of random matrices [67, 1, 72]. It should be noted that the exponent $a$ is an integer for $\beta=2,4$ for any number of channels $n_{1},n_{2}$, whereas for $\beta=1$ we require $|n_{1}-n_{2}|$ to be an odd integer. The distribution of the Landauer conductance follows as

[TABLE]

A simple scaling $y_{j}=T_{j}/g$ reveals that the above can be written in terms of the CDF of the largest eigenvalue of the fixed trace Wishart-Laguerre ensemble [20],

[TABLE]

[TABLE]

Plugging in the expression for $Q_{F}$ from (9) gives

[TABLE]

This equation provides the sought exact Landauer conductance distribution for general non-negative integer $a$, $\beta$. Corresponding Mathematica code is provided in the supplementary material. In Fig. 2, as examples, we show the distribution of scaled conductance, viz., $nP_{g}(ng)$ vs. $ng$ for $\beta=1,2,4$ and various $n,m$ combinations. The numerical results obtained using simulation of scattering matrices from circular ensembles are overlaid as symbols on the solid curves based on analytical results. We can see a perfect agreement.

5 Summary and conclusion

In this work, we have provided an efficient recursive scheme for the exact computation of the largest eigenvalue distribution of the integer $\beta$-Wishart-Laguerre ensemble. Applications to multiple channel communication, bipartite entanglement problems, and an exact solution to the Landauer conductance distribution in a chaotic mesoscopic cavity with ideal leads have been highlighted.

We would like to conclude by emphasising that the recursive approach proposed in this work for the Wishart-Laguerre largest eigenvalue can be obtained from the more general case of the Jacobi ensemble [42, 43]. The details of this, and its consequences, are planned for a future work.

Acknowledgements

P.J.F. acknowledges support from the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS) and ARC grant DP170102028. S.K. acknowledges the support by the grant EMR/2016/000823 provided by SERB, DST, Government of India. Comments by Mario Kieburg on an earlier draft of the manuscript are appreciated.

Appendix A Refinement of upper limit of inner summation in equation (6)

We discuss here the refinement of the upper limit of inner summation in (6). For this we consider the $j$-point correlation function,

[TABLE]

and the following expansion of $Q(x)$ in terms of $\rho_{(j)}$:

[TABLE]

We then use the fact that for large $x_{1},\dots,x_{j}$, $\rho_{(j)}(x_{1},\dots,x_{j})$ is proportional to

[TABLE]

and perform integration by parts multiple times. The latter procedure reveals, for each $j$, the exponent $\kappa$ in the leading large $x$ form $x^{\kappa}e^{-\beta jx/2}$ and thus the upper limit in the summation over $k$ in (6). The value $\kappa=ja+j(n-j)\beta$ is therefore obtained, as appears in (6).