Random Matrix Models, Double-Time Painlevé Equations, and Wireless Relaying
aa r X i v : . [ n li n . S I] J un J. Math. Phys. , 063506 (2013) Random Matrix Models, Double-Time Painlev´e Equations, and Wireless Relaying
Yang Chen, a) Nazmus S. Haq, b) and Matthew R. McKay c)1) Faculty of Science and Technology, Department of Mathematics, University of Macau,Av. Padre Tom´as Pereira, Taipa Macau, China Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2BZ,UK Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology(HKUST), Clear Water Bay, Kowloon, Hong Kong (Dated: 22 December 2012; Revised: 26 May 2013)
This paper gives an in-depth study of a multiple-antenna wireless communication scenarioin which a weak signal received at an intermediate relay station is amplified and then for-warded to the final destination. The key quantity determining system performance is thestatistical properties of the signal-to-noise ratio (SNR) g at the destination. Under certainassumptions on the encoding structure, recent work has characterized the SNR distributionthrough its moment generating function, in terms of a certain Hankel determinant gener-ated via a deformed Laguerre weight. Here, we employ two different methods to describethe Hankel determinant. First, we make use of ladder operators satisfied by orthogonalpolynomials to give an exact characterization in terms of a “double-time” Painlev´e dif-ferential equation, which reduces to Painlev´e V under certain limits. Second, we employDyson’s Coulomb Fluid method to derive a closed form approximation for the Hankel de-terminant. The two characterizations are used to derive closed-form expressions for thecumulants of g , and to compute performance quantities of engineering interest.PACS numbers: 02.30.Ik, 89.70.-a, 02.10.YnKeywords: Orthogonal polynomials; MIMO systems; random matrix theory; Painlev´eequations a) Electronic mail: [email protected], [email protected] b) Electronic mail: [email protected]; Author to whom correspondence should be addressed. c) Electronic mail: [email protected] . INTRODUCTION Over the past decade, multiple-input multiple-output (MIMO) antenna systems employingspace-time coding have revolutionized the wireless industry. Such systems are well-known tooffer substantial benefits in terms of channel capacity, as well as improved diversity and link re-liability, and as such these techniques are being incorporated into a range of emerging industrystandards. More recently, relaying strategies have been also proposed as a means of further im-proving the performance of space-time coded MIMO networks. Such methods are particularlyeffective for improving the data transmission quality of users who are located near the peripheryof a communication cell.Various MIMO relaying strategies have been proposed in the literature; see, e.g., Refs. 1–6, of-fering different trade-offs in various factors such as performance, complexity, channel estimationrequirements, and feedback requirements. In Ref. 7, a communication technique was consideredwhich employed a form of orthogonal space-time block coding (OSTBC), along with non-coherentamplify-and-forward (AF) processing at the multi-antenna relay. This method has the advantageof operating with only low complexity, requiring only linear processing at all terminals, achievinghigh diversity, and not requiring any short-term channel information at either the relays or thetransmitter. For this system, an important performance measure—the so-called outage probabil-ity —which gives a fundamental probabilistic performance measure over random communicationchannels, was considered in Refs. 7 and 8, based on deriving an expression for the moment gen-erating function of the received signal to noise ratio (SNR). The outage probability could then becomputed by performing a subsequent numerical Laplace Transform inversion. The exact momentgenerating function results presented in Refs. 7 and 8, however, are quite complicated functionsinvolving a particular
Hankel determinant, and they yield little insight. Moreover, for all but smallsystem configurations, the expressions are somewhat difficult to compute, particularly when im-plementing the Laplace inversion.In this paper, we significantly expand and elaborate upon the results of Refs. 7 and 8 by employ-ing analytical tools from random matrix theory and statistical physics. Technically, the challengeis to appropriately characterize the Hankel determinant which arises in the moment generatingfunction computation. In general, the Hankel determinant takes the form, D n = det (cid:0) m i + j (cid:1) n − i , j = , (1)2enerated from the moments of a certain weight function w ( x ) , 0 ≤ x < ¥ , m k : = ¥ Z x k w ( x ) dx , k = , , , . . . . Note that the above is merely one of several possible representations for D n , with equivalent formsregularly found in the fields of statistical physics and random matrix theory.For the problem at hand, we are faced with a Hankel determinant that is generated from aweight function of the form, w AF ( x , T , t ) = x a e − x (cid:18) t + xT + x (cid:19) N s , ≤ x < ¥ , (2)which is a “two-time” deformation of the classical Laguerre weight, x a e − x . The parameters in thisweight, satisfy the following conditions: a > − , T : = t + cs , t > , c > , N s > , ≤ s < ¥ . (3)For our problem both a and N s are integers, where a : = | N R − N D | is the absolute differencebetween the number of relay and destination antennas, N R and N D respectively. The parameters a and N s are determined by the model, but in fact can be extended to take non-integer values suchthat a > − N s > We will apply two different methods to characterize this Hankel determinant. For the firstmethod, we will derive new exact expressions for D n [ w AF ] by employing the theory of orthogonalpolynomials associated with the weight w AF ( x , T , t ) and their corresponding ladder operators. Forsuch methods, extensive literature exists (see, e.g., Refs. 10–20 for their use in applications involv-ing unitary matrix ensembles), though in the context of information theory and communicationsthe techniques have only very recently been introduced by the first and third authors in Ref. 13.For the second method, we derive an approximation for the Hankel determinant using the gen-eral linear statistics theorem obtained in Ref. 21, based on Dyson’s Coulomb Fluid models. These results are essentially the Hankel analogue of asymptotic results for Toeplitz determinants. Once again, these techniques have been used extensively, particularly in statistical physics, thoughonly very recently have they been applied to address problems in communications and informationtheory.
It is important to note that in addition to the two methodologies advocated above, there ex-ists other integrable systems approaches which can be used for characterizing Hankel determi-nants. For example, D n may be written in an equivalent matrix integral formulation, from which3 ‘deform-and-study’ or isomonodromic deformation approach may be adopted. Essentially, thisidea involves embedding D n into a more general theory of the t -function, using bilinear iden-tities and linear Virasoro constraints. For details, see Refs. 30–32. Yet another approach is tocharacterize D n through the use of Fredholm determinants as employed by Tracy and Widom. Both these exact, non-perturbative integrable systems methods have their own specific advantagesand disadvantages, and in most cases lead to non-linear ordinary differential or partial differen-tial equations (ODE/PDEs) satisfied by the Hankel determinant. However, these equations areusually of higher order (equations of Chazy type usually appear), from which first integrals haveto be found to reduce to a second order ODE. The advantage of the ladder operator method isthat closed-form second order equations are directly obtained for a quantity related to the Hankeldeterminant, bypassing the need to find a first integral, or evaluate any multiple integral.The rest of this paper is organized as follows. In Sections I A and I B, we present a brief dis-cussion of the model which underpins the MIMO-AF wireless communication system of interest,and some basic measures which are employed to quantify system performance. Then, in SectionsI C and I D, we go on to introduce the moment generating function and cumulants of interest, andpose the key mathematical problems to be dealt with in the remainder of the paper.In Section II, we establish an exact finite n characterization of the Hankel determinant D n ,employing the theory of orthogonal polynomials and their ladder operators.In Section III, we introduce the Coulomb Fluid method, where we first give a brief overview ofits key elements, following Refs. 25, 33, 13 and 21. In Section III B, we compute an asymptotic(large n ) approximation for the Hankel determinant D n , and thus a corresponding characterizationfor the moment generating function of interest. In Section III C, the Coulomb Fluid representationfor the moment generating function of the received SNR is shown to yield extremely accurateapproximations for the error performance of OSTBC MIMO systems with AF relaying, even whenthe system dimensions are particularly small.We also employ our analytical results to compute closed-form expressions for the cumulantsof the received SNR, first via the Coulomb Fluid method in Section III D, and then presenting arefined analysis based on Painlev´e equations in Sections IV and V.Subsequently, we give an asymptotic characterization of the moment generating function, validfor scenarios for which the average received SNR is high, deriving key quantities of interest tocommunication engineers, including the so-called diversity order and array gain. These resultsare, once again, established via the Coulomb Fluid approximation in Section VI, and subsequently4alidated with the help of a Painlev´e characterization in Section VII. A. Amplify and Forward Wireless Relay Model
Here we briefly recall the background for the model, having been developed in Ref. 7. Thedual hop MIMO communication system features a source, relay and destination terminal, having N s , N R and N D antennas respectively. In the process of transmitting a signal, each transmissionperiod is divided into two time slots. In the first time slot, the source transmits to the relay. Therelay then amplifies its received signal subject to an average power constraint, prior to transmittingthe amplified signal to the destination terminal during the second time-slot. We assume that thesource and destination terminals are sufficiently separated such that the direct link between themis negligible (i.e., all communication is done via the relay).Let H ∈ C N R × N s and H ∈ C N D × N R represent the channel matrices between the source andrelay, and the relay and destination terminals respectively. Each channel matrix is assumed tohave uncorrelated elements distributed as C N ( , ) . The destination is assumed to have perfectknowledge of H and either H or the cascaded channel H H , while the relay and source terminalshave no knowledge of these.A situation is considered where the source terminal transmits data using a method called OS-TBC encoding. In this situation, groups of independent and identically distributed complexGaussian random variables (referred to as information “symbols”) s i , i = , . . . , N are assignedvia a special codeword mapping to a row orthogonal matrix X = ( x , . . . , x N P ) ∈ C N s × N P satisfyingthe power constraint E (cid:2) k x k k (cid:3) = ¯ g , where N P is the number of symbol periods used to send eachcodeword.Since it takes N P symbol periods to transmit N symbols, the coding rate is then defined as R = NN P . (4)The received signal matrix at the relay terminal at the end of the first time slot, Y = ( y , . . . , y N P ) ∈ C N R × N P , is given by Y = H X + N , (5) The notation
C N ( m , s ) represents a complex Gaussian distribution with mean m and variance s . N ∈ C N R × N P has uncorrelated entries distributed as C N ( , ) , representing normalizednoise samples at the relay terminal. The relay then amplifies the signal it has received by a constantgain matrix G = ˜ a I N R , where ˜ a = ˜ b ( + ¯ g ) N R . (6)In the above, ˜ b is the total power constraint imposed at the relay, i.e., E (cid:2) k Gy k k (cid:3) ≤ ˜ b , whilst ¯ g rep-resents the average received SNR at the relay. The received signal R ∈ C N D × N P at the destinationterminal at the end of the second time slot is then given by R = ˜ a H Y + W = ˜ a H H X + ˜ a H N + W , (7)where W ∈ C N D × N P has uncorrelated entries distributed as C N ( , ) , representing normalizednoise samples at the destination terminal.Next, the receiver applies the linear (noise whitening) operation,˜ R = K − / R , K : = ˜ a H H †2 + I N D , (8)to yield the equivalent input-output model ˜ R = ˜ HX + ˜ N , (9)where ˜ H = ˜ a K − / H H and ˜ N ∈ C N D × N P has uncorrelated entries distributed as C N ( , ) .Based on the above relationship, standard linear OSTBC decoding can be applied (see, e.g.,Ref. 34). This results in decomposing the matrix model (9) into a set of parallel non-interactingsingle-input single-output relationships given by˜ s l = || ˜ H || F s l + h l , l = , . . . , N , where h l is distributed as C N ( , ) , with || ˜ H || F representing the Frobenius norm (or matrixnorm) of ˜ H .The quantity that we are interested in, the instantaneous SNR for the l th symbol g l , can then bewritten as g l = || ˜ H || F E (cid:2) | s l | (cid:3) = ¯ g ˜ bRN s ( + ¯ g ) N R Tr (cid:16) H †1 H †2 K − H H (cid:17) . (10)Since the right-hand side is independent of l , we may drop the l subscript and denote the instanta-neous SNR as g without loss of generality. 6 . Wireless Communication Performance Measures Here we recall some basic measures which are employed to quantify the performance of wire-less communication systems. One of the most common measures is the so-called symbol errorrate (SER), which quantifies the rate in which the transmitted symbols (or signals) are detectedincorrectly at the receiver.For signals which are designed using standard
M-ary phase shift keying (MPSK) digital mod-ulation formats, the phase of a transmitted signal is varied to convey information, where M ∈{ , , , , . . . } represents the number of possible signal phases. The SER can be expressed asfollows P MPSK = p Q Z M g (cid:18) g MPSK sin q (cid:19) d q , (11)where M g ( · ) is the moment generating function of the instantaneous SNR g at the receiver, and Q = p ( M − ) / M and g MPSK = sin ( p / M ) are modulation-specific constants. As also presentedin Ref. 36, the SER may be approximated in terms of M g ( · ) but without the integral, via thefollowing expression: P MPSK ≈ (cid:18) Q p − (cid:19) M g ( g MPSK ) + M g (cid:18) g MPSK (cid:19) + (cid:18) Q p − (cid:19) M g (cid:18) g MPSK sin Q (cid:19) . (12)In addition to the SER, another useful quantity is the so-called amount of fading (AoF), whichserves to quantify the degree of fading (i.e., the level of randomness) in the wireless channel, andis expressed directly in terms of the cumulants of g . Specifically, the AoF is defined as follows:AoF = k / k , (13)where k and k are the mean and variance of g respectively. As shown in Ref. 37, for example,this quantity can be used to describe the achievable capacity of the wireless link when the averageSNR is low.For our AF relaying system under consideration, it is clear that a major challenge is to charac-terize the moment generating function and also the cumulants of the instantaneous SNR given in(10). This is the key focus of the paper. 7 . Statistical Characterization of the SNR g Here we introduce the moment generating function and cumulants of interest, and pose the keymathematical problems to be dealt with in the remainder of the paper.Defining the positive integers, m : = max ( N R , N D ) , n = min ( N R , N D ) , (14)where n = M g ( s ) = n ! R [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x m − nk e − x k (cid:18) + ˜ a x k + ˜ a ( + ¯ g sRNs ) x k (cid:19) N s dx k n ! R [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x m − nk e − x k dx k . (15)From the above, we may then compute the l th cumulant of g via k l = ( − ) l d l ds l log M g ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = . (16)Let a : = m − n , (17) t : = a , (18) c : = ¯ g RN s , (19) T = T ( s ) : = t ( + cs ) . (20)We now consider the moment generating function (15) as a function of two variables, ( T , t ) , or ( s , t ) , since T = t / ( + cs ) . We may then write the moment generating function (15) as M g ( T , t ) = (cid:18) Tt (cid:19) nN s n ! R [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x a k e − x k (cid:16) t + x k T + x k (cid:17) N s dx k n ! R [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x a k e − x k dx k . (21)We see that this involves the weight with the parameters T and tw AF ( x , T , t ) = x a e − x (cid:18) t + xT + x (cid:19) N s , (22)8hich is a deformation of the classical generalized Laguerre weight w ( a ) Lag ( x ) = x a e − x , ≤ x < ¥ . (23)Note that w AF ( x , t , t ) = w ( a ) Lag ( x ) . The multiple integral representation (21) is expressed as a ratio of Hankel determinants: M g ( T , t ) = (cid:18) Tt (cid:19) nN s D n [ w AF ( · , T , t )] D n [ w ( a ) Lag ( · )] , (24) = (cid:18) Tt (cid:19) nN s det (cid:16) m i + j ( T , t ) (cid:17) n − i , j = det (cid:16) m i + j ( t , t ) (cid:17) n − i , j = , (25)where m j ( T , t ) is the j th moment of the weight m j ( T , t ) = ¥ Z x j w AF ( x , T , t ) dx , j = , , , . . . For N s ∈ N , the moments of the weight w AF are expressed in terms of the Kummer function of thesecond kind, m j ( T , t ) = t a + j + G ( a + j + ) N s (cid:229) k = (cid:18) N s k (cid:19) (cid:18) t − TT (cid:19) k U ( a + j + , a + j + − k , T ) . (26)The Hankel determinant for T = t , namely, D n [ w AF ( · , t , t )] = D n [ w ( a ) Lag ( · )] , can be regarded as a normalization constant, so that M g ( t , t ) =
1, and its closed form expressionis well-known (see Ref. 38). D n [ w AF ( · , t , t )] = n − (cid:213) i = G ( a + i + ) G ( i + ) , = G ( n + ) G ( n + a + ) G ( a + ) , (27)where G ( z ) is the Barnes G-function defined by G ( z + ) = G ( z ) G ( z ) with G ( ) = k l = c l (cid:18) T t ddT (cid:19) l log M g ( T , t ) (cid:12)(cid:12)(cid:12)(cid:12) T = t . (28)9ote that an equivalent expression to the result given in (26) has been previously reported inRef. 7. This result, whilst having numerical computation advantages when the system dimensionsare small, has a number of shortcomings. Most importantly, its complexity does not lead to usefulinsight into the characteristics of the probability distribution of the SNR g , and it does not clearlyreveal the dependence on the system parameters. Moreover, in this form, the expression is notamenable to asymptotic analysis, e.g., as the number of antennas grow sufficiently large, and insuch cases its numerical computation becomes unwieldy.In the following, we will seek alternative simplified representations with the aim of overcomingthese shortcomings. The key challenge is to characterize the Hankel determinant in (24). Remark 1.
It is of interest to mention here that for the special case a = , the Hankel determinant(24) is directly related to the shot-noise moment generating function of a disordered quantumconductor. This was investigated in Ref. 39 using the Coulomb Fluid method. Specifically, the twomoment generating functions are related by M g ( T , t ) | a = = M Shot-Noise ( T , z ) z nN s , (29) where z : = t / T and M Shot-Noise ( T , z ) is the Hankel determinant generated viaw ( x , T , z ) = e − T x (cid:18) z + x + x (cid:19) N s , ≤ x < ¥ . (30) Of course, T , z and N s have different interpretations in this situation.We mention here that the operator theory approach of Ref. 40 justifies the Coulomb Fluid resultsobtained in Ref. 41 on the shot-noise problem. D. Alternative Characterization of the SNR g An alternative characterization for the moment generating function that will also prove to beuseful is derived as follows. From the definitions of m and n , there are two possible choices forthe parameter t , t = a = ( + ¯ g ) N R ˜ b = ( + ¯ g ) n ˜ b N R ≤ N D , ( + ¯ g ) nr ˜ b , where r : = m / n , N R > N D . (31)10e first consider the sub-case N R ≤ N D . To this end, starting with (21) and with the change ofvariables, x i → nx i , i = , . . . , n , we obtain M g ( T ′ , t ′ ) = (cid:18) T ′ t ′ (cid:19) nN s n ! R [ , ¥ ) n n (cid:213) l = dx l exp " (cid:229) ≤ j < k ≤ n log | x j − x k | − n n (cid:229) j = (cid:0) x j − b log x j (cid:1) − n (cid:229) j = N s log (cid:16) T ′ + x j t ′ + x j (cid:17) n ! R [ , ¥ ) n n (cid:213) l = dx l exp " (cid:229) ≤ j < k ≤ n log | x j − x k | − n n (cid:229) j = (cid:0) x j − b log x j (cid:1) , (32) where t ′ : = tn , (33) T ′ : = Tn , (34) b : = mn − . (35)Note that in terms of s , T ′ can be written as T ′ = t ′ + cs . (36)Equivalently, we can write (32) as M g ( T ′ , t ′ ) = (cid:18) T ′ t ′ (cid:19) nN s Z n ( T ′ , t ′ ) Z n ( t ′ , t ′ ) , (37)where Z n ( T ′ , t ′ ) : = D n (cid:0) nT ′ , nt ′ (cid:1) = n ! Z [ , ¥ ) n exp " − F ( x , . . . , x n ) − n (cid:229) j = N s log (cid:18) T ′ + x j t ′ + x j (cid:19) n (cid:213) l = dx l , (38) and F ( x , . . . , x n ) : = − (cid:229) ≤ j < k ≤ n log | x j − x k | + n n (cid:229) j = (cid:0) x j − b log x j (cid:1) . (39) Remark 2.
We introduce the variables T ′ and t ′ in order to account for the n-dependence of thevariables T and t. This is important since the above representation will be useful for derivinga large n approximation for the moment generating function based on the Coulomb Fluid linearstatistics approach in Section III. For the sake of brevity, instead of defining a new function, we write M g ( T ′ , t ′ ) in place of M g ( nT ′ , nt ′ ) . N R > N D , we have from (31) that t is instead given by t = ( + ¯ g )( + b ) n ˜ b , (40)where b = mn − . Hence equations (37)–(39) and the results of Sections III-V are valid for thecase N R > N D upon transforming t ′ and T ′ to t † and T † via t ′ = : ( + b ) t † (41)and T ′ = : ( + b ) T † (42)respectively. In this case, we may write T † in terms of the variable s as T † = t † + cs . (43) II. PAINLEV ´E CHARACTERIZATION VIA THE LADDER OPERATORFRAMEWORK
Using the ladder operator framework, and treating T and t as independent variables, the resultbelow is proved in Appendix A. This gives a PDE satisfied by log M g ( T , t ) in the variables T and t . Theorem 1.
Let the quantity H n ( T , t ) be defined through the Hankel determinant D n ( T , t ) asH n ( T , t ) : = ( T ¶ T + t ¶ t ) log D n ( T , t ) (44) = ( T ¶ T + t ¶ t ) log M g ( T , t ) , (45) where the second equality follows from (24). Then H n ( T , t ) satisfies the following PDE: :H n = − ( ¶ T H n )( ¶ t H n ) + ( n − N s + a + T ) ¶ T H n + ( n + N s + a + t ) ¶ t H n ± A ( H n ) ∓ A ( H n ) − h ( T ¶ T T + t ¶ Tt ) H n ih ( T ¶ Tt + t ¶ tt ) H n i + A ( H n ) A ( H n ) h T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) i , (46) Dropping the T and t dependence notation for the sake of brevity. here { A ( H n ) } = (cid:16) ( T ¶ TT + t ¶ Tt ) H n (cid:17) + (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17)(cid:16) ¶ T H n (cid:17)(cid:16) ¶ T H n − N s (cid:17) , (47) and { A ( H n ) } = (cid:16) ( T ¶ Tt + t ¶ tt ) H n (cid:17) + (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17)(cid:16) ¶ t H n (cid:17)(cid:16) ¶ t H n + N s (cid:17) . (48) Remark 3.
If H n ( T , t ) is a function of T only, i.e., H n ( T , t ) = Y n ( T ) , then the PDE (46) reduces tothe Jimbo-Miwa-Okamoto s -form associated with Painlev´e V: (cid:16) TY n ′′ (cid:17) = (cid:16) Y n − TY n ′ + ( Y n ′ ) + ( n + n + n + n ) Y n ′ (cid:17) − (cid:0) n + Y n ′ (cid:1)(cid:0) n + Y n ′ (cid:1)(cid:0) n + Y n ′ (cid:1)(cid:0) n + Y n ′ (cid:1) , where ′ denotes differentiation with T , and n = , n = − n − a , n = − n , n = − N s . Such a reduction can be obtained from a t → ¥ limit in (21). Remark 4.
A similar reduction can be found when H n ( T , t ) is a function of t only, i.e., H n ( T , t ) = Y n ( t ) , which is a s − form of Painlev´e V with parameters n = , n = − n − a , n = − n , n = N s . This can be obtained from a T → ¥ limit in (21). With a change of variables the PDE, equation (46), can be converted into a form which will beconvenient for the later computation of cumulants. Specifically, let T = v + cs , (49) t = v . (50)Under this transformation, H n ( T , t ) becomes H n (cid:16) v + cs , v (cid:17) , which we write as H n ( s , v ) for the sakeof brevity. Hence, H n ( s , v ) = v ¶ v log M g ( s , v ) . (51)13e have that v = t and s = c ( tT − ) and consequently, ¶¶ T = − ( + cs ) vc ¶¶ s , (52) ¶¶ t = ( + cs ) vc ¶¶ s + ¶¶ v . (53)Under the change of variables, the original PDE (46) becomes the following PDE in the variables ( s , v ) : H n = ( + cs ) vc (cid:16) ¶ s H n (cid:17)(cid:16) ¶ v H n (cid:17) + ( + cs ) ( vc ) (cid:16) ¶ s H n (cid:17) + ( n + N s + a + v ) (cid:16) ¶ v H n (cid:17) + N s ( + cs ) vc (cid:16) ¶ s H n (cid:17) − ( n − N s + a ) s ( + cs ) v (cid:16) ¶ s H n (cid:17) ± A ∗ ( H n ) ∓ A ∗ ( H n ) − A ∗ ( H n ) A ∗ ( H n )( vc ) − ( + cs ) h ( v ¶ vs − ¶ s ) H n ih ( + cs ) (cid:0) v ¶ vs − ¶ s (cid:1) H n + v c (cid:0) ¶ vv H n (cid:1)i ( vc ) h v ( ¶ v H n ) − H n + n ( n + a ) i , (54) where A ∗ ( H n ) and A ∗ ( H n ) obtained from a corresponding transformation are ( vc ) A ∗ ( H n ) = ( + cs ) h(cid:0) v ¶ vs − ¶ s (cid:1) H n i + ( + cs ) (cid:20) v (cid:0) ¶ v H n (cid:1) − H n + n ( n + a ) (cid:21)(cid:20) ¶ s H n (cid:21)(cid:20) ( + cs ) (cid:0) ¶ s H n (cid:1) + N s vc (cid:21) , (55) and ( vc ) A ∗ ( H n ) = h ( + cs ) (cid:0) v ¶ vs − ¶ s (cid:1) H n + v c (cid:0) ¶ vv H n (cid:1)i + (cid:20) v (cid:0) ¶ v H n (cid:1) − H n + n ( n + a ) (cid:21)(cid:20)(cid:16) ( + cs ) ¶ s + vc ¶ v (cid:17) H n (cid:21) × (cid:20)(cid:16) ( + cs ) ¶ s + vc ¶ v (cid:17) H n + N s vc (cid:21) . (56) We note that in the PDE (54), the only higher order derivatives are the mixed derivatives withrespect to s and v , and second order derivatives with respect to v .It is of interest to note that the Hankel determinant in (21) or (24) has the following asymptoticforms: lim T → ¥ (cid:18) T nN s D n [ w AF ( · , T , t )] (cid:19) = D n [ w dLag ( · , t , a , N s )] , (57)lim t → ¥ (cid:18) t − nN s D n [ w AF ( · , T , t )] (cid:19) = D n [ w dLag ( · , T , a , − N s )] , (58)14here the generating weight is the “first-time” deformation of the Laguerre weight w dLag ( x , T , a , N s ) = x a e − x ( x + T ) N s . (59)In previous work by the authors, the Hankel determinant D n [ w dLag ( · , T , a , N s )] was shown toarise in the analysis of the moment generating function of the single-user MIMO channel capacity,and was shown to involve a Painlev´e V differential equation. III. COULOMB FLUID METHOD FOR LARGE n ANALYSIS
In this section, we make use of the Coulomb Fluid method, which is particularly convenientwhen the size of the matrix n is large, to describe the MIMO-AF problem. The idea is to treatthe eigenvalues as identically charged particles, with logarithmic repulsion, and held together byan external potential. When n , in this context the number of particles, is large, this assemblyis regarded as a continuous fluid, first put forward by Dyson, where the eigenvalues weresupported on the unit circle. For a detailed description of cases where the charged particles aresupported on the line, see Refs. 33, 21 and 25.An extension of the methodology to the study of linear statistics, namely, the sum of functionsof the eigenvalues of the form n (cid:229) j = f ( x j ) , can be found in Ref. 21 and will be used extensively in this paper. The main benefit of thisapproach, based on singular integral equations, is that it leads to relatively simple expressions forcharacterizing our moment generating function.In Section III A, for completeness, we give a brief overview of the key elements of the CoulombFluid method, following Refs. 33, 21, 13 and 25. In Section III B, we compute explicit solutionsfor the key quantities of interest in the Coulomb Fluid framework. Subsequently, in Section III C,we combine our Coulomb Fluid results with either (11) or (12) to directly yield analytical approx-imations for the SER of the MIMO-AF scheme under consideration. Quite remarkably, these areshown to be extremely accurate, even for very small dimensions. Similar accuracy is demonstratedin Section III D for SNR cumulant approximations obtained from the Coulomb Fluid results.15 . Preliminaries of the Coulomb Fluid Method We start by considering the ratio in the moment generating function expression (37), notingthat it is of the form Z n ( T ′ , t ′ ) Z n ( t ′ , t ′ ) = e − (cid:2) F n ( T ′ , t ′ ) − F n ( t ′ , t ′ ) (cid:3) , (60)where Z n ( T ′ , t ′ ) : = n ! Z [ , ¥ ) n exp " − F ( x , . . . , x n ) − n (cid:229) j = f ( x j , T ′ , t ′ ) n (cid:213) l = dx l (61) F ( x , . . . , x n ) : = − (cid:229) ≤ j < k ≤ n log | x j − x k | + n n (cid:229) j = v ( x j ) (62)and F n : = − log Z n is known as the free energy .This expression embraces the moment generating function expression (37) with appropriateselection of the functions v ( x ) and f ( x , T ′ , t ′ ) . A key motivation for writing our problem in thisform is that it admits a very intuitive interpretation in terms of statistical physics, as originallyobserved by Dyson (see Refs. 22–24). Specifically, if the eigenvalues x , . . . , x n are interpretedas the positions of n identically charged particles, then F ( x , . . . , x n ) is recognized as the totalenergy of the repelling charged particles, which are confined by the external potential n v ( x ) . Thefunction f ( x , T ′ , t ′ ) acts as a perturbation to the system, resulting in a modification to the externalpotential. The quantity F n ( T ′ , t ′ ) may be interpreted as the free energy of the system under anexternal perturbation f ( x , T ′ , t ′ ) , with F n ( t ′ , t ′ ) the free energy of the unperturbed system.For sufficiently large n , the system of particles, following Dyson, may be approximated as acontinuous fluid where techniques of macroscopic physics and electrostatics can be applied. Forlarge n we expect the external potential n v ( x ) to be strong enough to overcome the logarithmicrepulsion between the particles (or eigenvalues), and hence the particles or fluid will be confinedwithin a finite interval to be determined through a minimization process. For this continuous fluid,we introduce a macroscopic density s ( x ) dx , referred to as the equilibrium density. Since v ( x ) isconvex for x ∈ R , this density is supported on a single interval denoted by ( a , b ) , to be determinedlater (see Ref. 33 for a detailed explanation). The equilibrium density is obtained by minimizingthe free-energy functional: F n ( T ′ , t ′ ) : = b Z a s ( x ) (cid:16) n v ( x ) + n f ( x , T ′ , t ′ ) (cid:17) dx − n b Z a b Z a s ( x ) log | x − y | s ( y ) dxdy , (63)16ubject to b Z a s ( x ) dx = . (64)With Frostman’s Lemma, (p. 65) the minimizing s ( x ) dx can be characterized through the inte-gral equation n v ( x ) + n f ( x , T ′ , t ′ ) − n b Z a log | x − y | s ( y ) dy = A , (65)where x ∈ [ a , b ] and A is the Lagrange multiplier for the normalization condition (64), which canbe interpreted as the chemical potential of the fluid. Noting that the integral equation abovehas a logarithmic kernel, taking a derivative with respect to x ∈ ( a , b ) converts it into a singularintegral equation of the form v ′ ( x ) + f ′ ( x , T ′ , t ′ ) n = P b Z a s ( y ) x − y dy , (66)where P denotes Cauchy principal value.Noting the structure (in n ) of the left-hand side of (66), it is clear that s ( · ) must take the generalform: s ( x ) = s ( x ) + s c ( x , T ′ , t ′ ) n , (67)where s ( x ) dx is the density of the original system in the absence of any perturbation, while s c ( x , T ′ , t ′ ) represents the deformation of s ( x ) caused by f ( x , T ′ , t ′ ) . Furthermore, to satisfy(64), we have b Z a s ( x ) dx = , b Z a s c ( x , T ′ , t ′ ) dx = . (68)Substituting (67) into (66), and comparing orders of n , we see that s ( x ) solves v ′ ( x ) = P b Z a s ( y ) x − y dy , (69)and s c ( x , T ′ , t ′ ) solves f ′ ( x , T ′ , t ′ ) = P b Z a s c ( y , T ′ , t ′ ) x − y dy . (70)17ollowing Ref. 33, where the choice for the solution for s has been extensively discussed basedon the theory described in Ref. 43; the solution to (69) subject to the boundary condition s ( a ) = s ( b ) = s ( x ) = p ( b − x )( x − a ) p b Z a v ′ ( x ) − v ′ ( y )( x − y ) p ( b − y )( y − a ) dy , (71)together with a supplementary condition, b Z a v ′ ( x ) p ( b − x )( x − a ) dx = . (72)The solution to (70) subject to b R a s c ( x , T ′ , t ′ ) dx = s c ( x , T ′ , t ′ ) = p p ( b − x )( x − a ) P b Z a p ( b − y )( y − a ) y − x f ′ ( y , T ′ , t ′ ) dy . (73)Finally, the normalization condition (64) becomes b Z a x v ′ ( x ) p ( b − x )( x − a ) dx = p . (74)The end points of the support of the density s ( x ) , a , and b , are determined by (72) and (74), andwill depend on parameters associated with v . For a description see Ref. 33. (We mention here arelated problem, namely, the probability that there is a gap in the spectrum of the random matrix,has been studied using the Coulomb Fluid approach in Refs. 44 and 45.)With the above results, for sufficiently large n , we may approximate the ratio (60) as (seeRef. 21 for more details) Z n ( T ′ , t ′ ) Z n ( t ′ , t ′ ) ≈ exp (cid:16) − S AF2 ( T ′ , t ′ ) − nS AF1 ( T ′ , t ′ ) (cid:17) , (75)where S AF1 ( T ′ , t ′ ) = b Z a s ( x ) f ( x , T ′ , t ′ ) dx , (76) S AF2 ( T ′ , t ′ ) = b Z a s c ( x , T ′ , t ′ ) f ( x , T ′ , t ′ ) dx . (77)In the sequel, we will find explicit solutions for these quantities.18 . Coulomb Fluid Calculations for the SNR Moment Generating Function Using (75), the moment generating function M g ( T ′ , t ′ ) in (37) takes the form M g ( T ′ , t ′ ) = (cid:18) T ′ t ′ (cid:19) nN s Z n ( T ′ , t ′ ) Z n ( t ′ , t ′ ) , ≈ exp (cid:18) − S AF2 ( T ′ , t ′ ) − n (cid:20) S AF1 ( T ′ , t ′ ) − N s log (cid:18) T ′ t ′ (cid:19)(cid:21)(cid:19) , (78)for which, comparing (38) and (39) with (61) and (62), the functions v ( x ) and f ( x , T ′ , t ′ ) areidentified as v ( x ) : = x − b log x , (79) f ( x , T ′ , t ′ ) : = N s log (cid:18) T ′ + xt ′ + x (cid:19) , (80)where v ( x ) given by (79) is convex.From here, the l -th cumulant can be extracted from the formula k l = c l T ′ t ′ ddT ′ ! l log M g ( T ′ , t ′ ) (cid:12)(cid:12)(cid:12)(cid:12) T ′ = t ′ . (81)The objective of the subsequent analysis is to evaluate the quantities S AF1 and S AF2 . As we shall see,we will need to solve numerous integrals which are not readily available. Thus, to aid the reader,we have compiled these integrals in Appendix B.We start by considering the end points of the support a and b . These are determined by equa-tions (72) and (74). With the integral identities (B1)-(B3) in Appendix B, we obtain, after a feweasy steps, ab = b , a + b = ( + b ) , (82)which leads to a = + b − p + b , b = + b + p + b . (83)The limiting density s ( x ) in (71) can be computed using the integral identity (B2) as s ( x ) = p ( b − x )( x − a ) p x , x ∈ ( a , b ) , (84)19hich is the Marˆcenko-Pastur law. Meanwhile, with the aid of the integral identity (B4) inAppendix B, s c ( x , T ′ , t ′ ) given by (73) reads s c ( x , T ′ , t ′ ) = N s p p ( b − x )( x − a ) p ( T ′ + a )( T ′ + b ) x + T ′ − p ( t ′ + a )( t ′ + b ) x + t ′ ! . (85)Using s ( x ) and invoking the integral identities (B5)-(B7) in Appendix B, gives S AF1 ( T ′ , t ′ ) = N s (cid:18) t ′ − T ′ + p ( T ′ + a )( T ′ + b ) − p ( t ′ + a )( t ′ + b ) (cid:19) + N s ( a + b ) log (cid:18) √ T ′ + a + √ T ′ + b √ t ′ + a + √ t ′ + b (cid:19) + N s √ ab (cid:16) √ ab + p ( t ′ + a )( t ′ + b ) (cid:17) − t ′ (cid:16) √ ab + p ( T ′ + a )( T ′ + b ) (cid:17) − T ′ + N s ( a + b ) (cid:16)p ( t ′ + a )( t ′ + b ) + t ′ (cid:17) − ab (cid:16)p ( T ′ + a )( T ′ + b ) + T ′ (cid:17) − ab + N s ( a + b ) (cid:18) T ′ t ′ (cid:19) . (86) Moreover, using s c ( x , T ′ , t ′ ) and invoking the integral identities (B2), (B8) and (B12) in AppendixB gives S AF2 ( T ′ , t ′ ) = N s p ( T ′ + a )( T ′ + b ) p ( t ′ + a )( t ′ + b ) (cid:0)p ( T ′ + a )( T ′ + b ) + p ( t ′ + a )( t ′ + b ) (cid:1) − ( T ′ − t ′ ) ! . (87) C. SER Performance Measure Analysis Based on Coulomb Fluid
Combining (78) with (83), (86), and (87) yields a closed-form asymptotic expression for themoment generating function of the instantaneous SNR in (10). This, in turn, combined with either(11) or (12), directly yields analytical approximations for the SER of the MIMO-AF scheme underconsideration. The accuracy of these approximations is confirmed in Fig. 1, where they are com-pared with simulation results generated by numerically computing the exact SER via Monte Carlomethods. Different antenna configurations are shown, as represented by the form ( N s , N R , N D ) .The curves labeled “Coulomb (Exact SER)” were generated by substituting our Coulomb Fluidapproximation into the exact expression of the SER (11), whilst the curves labeled “Coulomb20Approx SER)” are generated by substituting our Coulomb Fluid approximation into the approxi-mate expression for the SER (12).From these curves, it is remarkable that the Coulomb Fluid based approximations, derivedunder the assumption of large n , very accurately predict the SER even for small values of n . Thisis evident, for example, by examining the set of curves corresponding to the configuration (2, 3,2), for which n =
2. In fact, even for the extreme cases with n =
1, the Coulomb Fluid curves stillyield quite high accuracy.We note that the results in Fig. 1 are representative of those presented previously in Ref. 7, Fig.5. The key difference therein was that the analytic curves were based on substituting into either(11) or (12) an equivalent determinant form of the moment generating function to that given in (25)and (26). That representation, involving a determinant of a matrix with elements comprising sumsof Kummer functions, is far more complicated than the Coulomb Fluid representation, which is asimple algebraic equation involving only elementary functions. The fact that the Coulomb Fluidrepresentation yields accurate approximations for the SER for small as well as large numbers ofantennas makes it a useful analytical tool for studying the performance of arbitrary MIMO-AFsystems.
D. Coulomb Fluid Analysis of Large n Cumulants of SNR
In this subsection we use the Coulomb Fluid results to study the asymptotic cumulants of theSNR. We suppose b is fixed, and distinguish between two cases, b = b >
1. Case 1: b = N R = N D . Then from (83), a = b =
4, leading to S AF1 ( T ′ , t ′ ) − N s log (cid:18) T ′ t ′ (cid:19) = N s (cid:16) t ′ − T ′ + p T ′ ( T ′ + ) − p t ′ ( t ′ + ) (cid:17) + N s log √ T ′ t ′ + p t ′ ( T ′ + ) √ T ′ t ′ + p T ′ ( t ′ + ) ! , (88) and S AF2 ( T ′ , t ′ ) = N s √ T ′ t ′ p ( T ′ + )( t ′ + ) √ T ′ t ′ p ( T ′ + )( t ′ + ) + T ′ + t ′ + T ′ t ′ ! . (89)21 −4 −3 −2 −1 SNR (dB) S E R Coulomb (Exact SER)Coulomb (Approx SER)Simulation (2,1,1)(2,3,2) (2,2,1)(4,5,3)(4,6,5)
FIG. 1. Illustration of the SER versus average received SNR (at relay) ¯ g ; comparison of analysis andsimulations. Each set of curves represents a specific antenna configuration of the form ( N s , N R , N D ) . Forconfigurations with N s =
2, the full-rate Alamouti OSTBC is used (i.e., R = N s =
4, a rate 1 / R = / M =
4. The relay power ˜ b is assumed to scale with ¯ g by setting ˜ b = ¯ g . From the closed-form approximation of the moment generating function (78) the cumulantscan be extracted using (81).Alternatively, recall that the variable T ′ is related to s via T ′ = t ′ + cs , where c = ¯ g / ( RN s ) . Therefore we expand log M g ( t ′ / ( + cs ) , t ′ ) obtained from the Coulomb Fluidformalism (88) and (89) in a Taylor series about s =
0. From (78),log M g ( t ′ / ( + cs ) , t ′ ) = ¥ (cid:229) j = ( − ) j s j j ! k j ( t ′ ) , (90)where the first few cumulants are k ( t ′ ) cN s = n (cid:16) t ′ + − p t ′ + t ′ (cid:17) , (91)22 ( t ′ ) c N s = N s ( t ′ + ) + n ( t ′ + ) − n p t ′ + t ′ (cid:18) − t ′ + (cid:19) , (92) k ( t ′ ) c N s = N s ( t ′ + ) + n ( t ′ + ) − n p t ′ + t ′ (cid:18) − t ′ + − ( t ′ + ) (cid:19) , (93) k ( t ′ ) c N s = N s ( t ′ + ) + n ( t ′ + ) − n p t ′ + t ′ (cid:18) − t ′ + − ( t ′ + ) − ( t ′ + ) (cid:19) , (94) k ( t ′ ) c N s = N s ( t ′ + ) + n ( t ′ + ) − n p t ′ + t ′ (cid:18) − t ′ + − ( t ′ + ) − ( t ′ + ) − ( t ′ + ) (cid:19) , (95)and k ( t ′ ) c N s = N s ( t ′ + ) + n ( t ′ + ) − n p t ′ + t ′ (cid:18) − t ′ + − ( t ′ + ) − ( t ′ + ) − ( t ′ + ) − ( t ′ + ) (cid:19) . (96)
2. Case 2: b > a and b , are given by (83). Hence, going through thesame process, the first five cumulants are k CF1 ( t ′ ) cN s = n (cid:18) t ′ + + b − q ( t ′ + b ) + t ′ (cid:19) , (97) k CF2 ( t ′ ) c N s = N s ( + b ) t ′ (cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:18) t ′ + + b (cid:19) − n q ( t ′ + b ) + t ′ − b + ( b + ) t ′ (cid:16) ( t ′ + b ) + t ′ (cid:17) , (98) k CF3 ( t ′ ) c N s = N s ( + b ) (cid:0) b + ( b + ) t ′ (cid:1) t ′ (cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:0) t ′ + b + (cid:1) − n q ( t ′ + b ) + t ′ − (cid:0) + b + b + ( b + ) t ′ (cid:1) ( t ′ + b ) + t ′ + (cid:0) b + b + ( b + )( b + ) t ′ (cid:1)(cid:0) ( t ′ + b ) + t ′ (cid:1) ! , (99) CF4 ( t ′ ) c N s = N s t ′ ( + b ) (cid:16) (cid:0) b + b + (cid:1) t ′ + b ( b + ) t ′ + b (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:0) t ′ + b + (cid:1) − n q ( t ′ + b ) + t ′ + t ′ ( t ′ + + b )( t ′ + b ) + t ′ + t ′ ( t ′ − − b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ (cid:16) t ′ + ( + b ) t ′ − − b (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) ! , (100) and k CF5 ( t ′ ) c N s = N s t ′ ( + b ) (cid:16) (cid:0) b + b + (cid:1) t ′ + b ( + b ) t ′ + b (cid:17) (cid:0) ( + b ) t ′ + b (cid:1)(cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:0) t ′ + b + (cid:1) − n q ( t ′ + b ) + t ′ + t ′ ( t ′ − b − )( t ′ + b ) + t ′ − t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) t ′ + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) t ′ + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) − u (cid:16) t ′ + ( b + ) t ′ ( t ′ − ) + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) ! . (101) The accuracy of these approximations is demonstrated in Fig.2, where they are compared withsimulation results generated by numerically computing the exact cumulants via Monte Carlomethods. As before, different antenna configurations are shown, as represented by the form ( N s , N R , N D ) . From these curves, the accuracy of our Coulomb Fluid based approximations isquite remarkable, even for small values of n .We note that exact expressions for k and k (i.e., the mean and variance) were derived pre-viously in Ref. 7, Corollaries 1 and 2, and these were expressed in terms of summations of de-terminants (for the mean) as well as a rank-3 tensor (for the variance), each involving Kummerfunctions. Such results are obviously far more complicated than the Coulomb fluid based cumulantexpressions in (97)–(99), which involve just very simple algebraic functions.24 k SimulationCoulomb (4,5,3) (2,3,2) (2,2,1) 4 6 8 10 12 14 16 18 200200400600800100012001400160018002000 SNR (dB) k (2,2,1)(2,3,2)(4,5,3)8 10 12 14 16 18 2000.511.522.533.544.55 x 10 SNR (dB) k (2,3,2) (2,2,1)(4,5,3) FIG. 2. Illustration of the mean k , variance k , and third cumulant k of the received SNR at the destination g , as a function of the average received SNR at the relay ¯ g . In each case, the simulated cumulants arecompared with those obtained based on the Coulomb fluid approximation. As in Fig. 1, each set of curvesrepresents a specific antenna configuration of the form ( N s , N R , N D ) . For configurations with N s =
2, thefull-rate Alamouti OSTBC is used (i.e., R = N s =
4, a rate 1 / R = / b is assumed to scale with ¯ g by setting ˜ b = ¯ g . IV. POWER SERIES EXPANSION FOR H n ( s , v ) In this section we seek power series solutions for (54), to determine the cumulants. We beginby removing the square roots of the PDE and arrive at the following lemma.
Lemma 1.
Let K : = (cid:16) v ( ¶ v H n ) − H n + n ( n + a ) (cid:17) , (102) and L : = K (cid:20) H n − ( + cs ) vc (cid:16) ¶ s H n (cid:17)(cid:16) ¶ v H n (cid:17) − ( + cs ) ( vc ) (cid:16) ¶ s H n (cid:17) − ( n + N s + a + v ) ¶ v H n − N s ( + cs ) vc ¶ s H n + ( n − N s + a ) s ( + cs ) v ¶ s H n (cid:21) ( + cs ) ( vc ) h ( v ¶ vs − ¶ s ) H n ih ( + cs ) (cid:0) v ¶ vs − ¶ s (cid:1) H n + v c (cid:0) ¶ vv H n (cid:1)i , (103) then the PDE (54) may be rewritten in an equivalent square-root-free form, as K A (cid:0) L + A (cid:1) = (cid:16) L + K ( A − A ) − A A (cid:17) , (104) where A and A are given by (55) and (56) respectively. Now, recall from (51) that the function H n ( s , v ) is related to the moment generating function(21) through H n ( s , v ) = v ¶ v log M g ( s , v ) , (105)and note that the logarithm of the moment generating function has the expansion,log M g ( s , v ) = ¥ (cid:229) j = ( − ) j k j ( v ) j ! s j , (106)where k j ( v ) is the j th cumulant. Note that for the sake of brevity we write M g ( s , v ) in place of M g (cid:0) v + cs , v (cid:1) . Consequently, from (105), H n ( s , v ) has the following expansion in s , H n ( s , v ) = v ¥ (cid:229) j = ( − ) j c j N s a ′ j ( v ) j ! s j , (107)where ′ denotes differentiation with respect to v , and a j ( v ) is related to the j th cumulant k j ( v ) by a j ( v ) c j N s = k j ( v ) . (108) A. Analysis of k In this subsection, the computation of the mean k is presented. To this end, upon substituting(107) into the PDE (104), keeping the lowest power of s in the resulting expansion gives a highlynonlinear ODE expressed in the factored form Y ( ) Y ( ) = , (109)where Y ( ) : = (cid:0) a ′′ ( v ) (cid:1) " N s v (cid:0) a ′′ ( v ) (cid:1) + n ( n + a ) N s (cid:0) a ′ ( v ) (cid:1) − N s (cid:0) a ′ ( v ) (cid:1) − n ( n + a ) ! , (110) We introduce a j in order to make a clearer comparison to the Coulomb Fluid model and without having to exactlyspecify a value for the constant c , since it falls out of the subsequent expansion. Y ( ) : = (cid:20) v (cid:0) a ′′′ ( v ) (cid:1) + v (cid:0) a ′′ ( v ) (cid:1) + n ( n + a ) (cid:18) a ′ ( v ) − (cid:19)(cid:21) − ( n + v + a ) (cid:20) v (cid:0) a ′′ ( v ) (cid:1) + n ( n + a ) (cid:0) a ′ ( v ) (cid:1) (cid:18) a ′ ( v ) − (cid:19)(cid:21) . (111) Hence, (109) is equivalent to either Y ( ) = Y ( ) = Y ( ) = a ( v ) = c v + c where c and c are constants to be determined. The second is more complicated but can be succinctlywritten as a ( v ) = f ( c , N s , n , a ) v sin (cid:0) √ n √ n + a log v (cid:1) + g ( c , N s , n , a ) v cos (cid:0) √ n √ n + a log v (cid:1) + c + v , (112) where c and c are integration constants, and f and g are algebraic functions (we omit these forbrevity).Whilst these are valid mathematical solutions, it is clear that by taking n large, they differdrastically from that predicted by the Coulomb Fluid method and, for small n , they differ withthe solutions obtained in Ref. 7. This suggests that these are not the solutions of interest to ourproblem; thus, we set them aside and henceforth focus on Y ( ) = k j ( v ) / ( c j N s ) ,since the PDE (104) gives rise to a system of ODEs satisfied by k j ( v ) / ( c j N s ) . However, (104) isnot supplemented by any initial conditions from which one deduces the initial conditions at t ′ = t ′ =
0. Consequently, the Coulomb Fluid results are in fact leadingorder approximations to the exact results. Without the Coulomb Fluid analysis, each cumulant k j ( v ) / ( c j N s ) would carry unknown constants of integration.We now examine the equation Y ( ) = n region, which leads to an asymptoticcharacterization of a ( v ) and thus k . To proceed, we scale the variable v by v = nt ′ . Also, notethat a = n ( m / n − ) ≡ n b . We find that a ( t ′ ) satisfies the following ODE:0 = (cid:20) t ′ n (cid:0) a ′′′ ( t ′ ) (cid:1) + t ′ n (cid:0) a ′′ ( t ′ ) (cid:1) + n ( + b ) (cid:0) a ′ ( t ′ ) − n (cid:1) (cid:21) − ( t ′ + + b ) (cid:20) t ′ (cid:0) a ′′ ( t ′ ) (cid:1) + n ( + b ) (cid:0) a ′ ( t ′ ) (cid:1) (cid:0) a ′ ( t ′ ) − n (cid:1) (cid:21) . (113)27or large n , keeping the leading order term, we arrive at (cid:18) da dt ′ − n (cid:19) = n ( t ′ + + b ) ( t ′ + b ) + t ′ , (114)whose solutions are a ( t ′ ) = n (cid:18) t ′ ± q ( t ′ + b ) + t ′ (cid:19) + C where C is a constant of integration. We retain the solution that is bounded as t ′ → ¥ , a ( t ′ ) = n (cid:18) t ′ − q ( t ′ + b ) + t ′ (cid:19) + C . (115)Comparing this with the corresponding result (97) obtained from the Coulomb fluid approxima-tion, we see that k CF ( ) = cN s n , which suggests a ( ) = n and hence C = n ( + b / ) . Therefore,the first cumulant k to O ( n ) becomes k Large n1 ( t ′ ) cN s = a ( t ′ )= n (cid:18) + b + t ′ − q ( t ′ + b ) + t ′ (cid:19) . (116) B. Analysis of k Having characterized k , we now turn to the variance k . To this end, equating the coefficientsof the next lowest power of s to zero in the expansion of the PDE (104), results in an ODE whichcan be factored into the following form Y ( ) (cid:16) Y ( ) + Y ( ) + Y ( ) (cid:17) = . (117) The ODE Y ( ) =
0, makes a reappearance, which we set aside, leaving the ODE Y ( ) + Y ( ) + Y ( ) = , (118)where Y ( ) , Y ( ) and Y ( ) are given by Y ( ) N s = v (cid:18) a ′ − (cid:19) (cid:16) v ( n − v + a ) (cid:0) a ′′ (cid:1) + ( n + v + a ) n ( n + a ) a ′ (cid:0) a ′ − (cid:1)(cid:17) a ′′′ − v ( n + v + a ) (cid:0) a ′′ (cid:1) + v ( n − v + a ) (cid:18) a ′ − (cid:19) (cid:0) a ′′ (cid:1) − v "(cid:16) ( n + v + a ) (cid:0) ( a + v )( a − v ) + n ( n + a ) (cid:1) − n ( n + a )( n + a ) (cid:17)(cid:0) a ′ (cid:1) (cid:0) a ′ − (cid:1) − n ( n + a )( n − v + a ) a ′′ (cid:1) + ( n + v + a ) n ( n + a ) a ′ (cid:0) a ′ − (cid:1) × (cid:20) v (cid:18) a ′ − (cid:19) a ′′ + (cid:16) ( v + a ) + vn (cid:17) a ′ ( − a ′ ) + n ( n + a ) (cid:21) , (119) ( ) = " v ( n ( n + a ) − ) (cid:0) a ′′ (cid:1) − v (cid:16) va ′′′ + n ( n + a ) (cid:0) a ′ − (cid:1)(cid:17) a ′′ + v n ( n + a ) (cid:18) v (cid:18) a ′ − (cid:19) a ′′ + (cid:16) ( v + a )( n + v + a ) + n ( n + a − ) (cid:17)(cid:0) a ′ (cid:1) (cid:0) a ′ − (cid:1) + n ( n + a ) + (cid:0) a ′ − (cid:1) a ′ − a ′ (cid:19) a ′′′ + v (cid:18) n ( n + a ) − (cid:19) (cid:0) a ′′ (cid:1) − v a ′′′ (cid:0) a ′′ (cid:1) + v " v ( n + v + a ) a ′′ − (cid:18)(cid:16) ( n + v + a ) − n ( n + a ) (cid:17) n ( n + a ) − v ( n + a ) − a (cid:19) a ′ − (cid:16) n ( n + a ) + vn + ( v + a ) a (cid:17) a ′ + n ( n + a ) (cid:16) ( n + v + a ) − n ( n + a ) + (cid:17) a ′′ (cid:1) − n ( n + a ) v a ′′ (cid:18) a ′ − (cid:19) (cid:0) va ′′′ − a ′′ (cid:1) − vn ( n + a ) a ′′ , (120) and Y ( ) n ( n + a ) = v "(cid:16) ( n + v + a ) + n ( n + a ) − (cid:17)(cid:0) a ′ (cid:1) − (cid:16) ( n + v + a ) − n ( n + a ) + (cid:17) a ′ + (cid:18)(cid:16) ( n + v + a ) + − n ( n + a ) (cid:17) a ′ − ( n + v + a ) + + n ( n + a ) (cid:19) a ′ a ′′ + v (cid:20)(cid:16) nv + ( v + a ) (cid:17) (cid:0) a ′ (cid:1) ( − a ′ ) + n ( n + a ) (cid:21) a ′′ + (cid:16) nv + ( v + a ) a (cid:17) a ′ ( − a ′ ) a ′ + (cid:20) ( n + v + a ) (cid:16) n + a − ( n + v + a ) n ( n + a ) (cid:17) + n ( n + a ) (cid:0) n ( n + a ) − (cid:1)(cid:21) (cid:0) a ′ (cid:1) ( a ′ − )+ (cid:16) ( n + v + a ) − n ( n + a ) (cid:17) n ( n + a ) (cid:0) a ′ (cid:1) − n ( n + a ) (cid:18) − a ′ + n ( n + a ) (cid:19) + (cid:16) n ( n + a ) − ( n + v + a ) − (cid:17) n ( n + a ) a ′ . (121) Clearly, Y ( ) depends on a only, whilst Y ( ) and Y ( ) depend on both a and a . To extract thelarge n behavior of k , we replace v by nt ′ in (118) and make use of the large n formula for a ( t ′ ) in (116), resulting to a highly non-linear ODE satisfied by a ( t ′ ) . To proceed further, keeping thehighest powers of n , a first order equation is obtained for a ( t ′ ) , da ( t ′ ) dt ′ = n − n q ( t ′ + b ) + t ′ " t ′ + b + ( t ′ + b ) + t ′ − t ′ ( b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − N s t ′ ( b + )( t ′ − b )( t ′ + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) . (122) Integrating this leads to an expression for a ( t ′ ) which, again yields an unknown integrationconstant. This constant can be determined through a comparison with the Coulomb Fluid resultsfor k ( t ′ ) in (98) at t ′ = , giving a ( ) = n . 29he above analysis gives the leading order characterization of the variance, k Large n2 ( t ′ ) c N s = a ( t ′ )= n (cid:18) t ′ + + b (cid:19) − n q ( t ′ + b ) + t ′ − b + ( b + ) t ′ (cid:16) ( t ′ + b ) + t ′ (cid:17) + N s ( + b ) t ′ (cid:0) ( t ′ + b ) + t ′ (cid:1) . (123) C. Beyond k and k The same procedure for computing k and k easily extends to higher cumulants. Here we givethe leading order formula for k , k and k , k Large n3 ( t ′ ) c N s = a ( t ′ )= n (cid:0) t ′ + b + (cid:1) + N s t ′ ( + b ) (cid:0) b + ( b + ) t ′ (cid:1)(cid:0) ( t ′ + b ) + t ′ (cid:1) , − n q ( t ′ + b ) + t ′ " − (cid:0) + b + b + ( b + ) t ′ (cid:1) ( t ′ + b ) + t ′ + (cid:0) b + b + ( b + )( b + ) t ′ (cid:1)(cid:0) ( t ′ + b ) + t ′ (cid:1) (124) k Large n4 ( t ′ ) c N s = a ( t ′ )= − n q ( t ′ + b ) + t ′ × + t ′ ( t ′ + + b )( t ′ + b ) + t ′ + t ′ ( t ′ − − b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ (cid:16) t ′ + ( + b ) t ′ − − b (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:0) t ′ + b + (cid:1) + N s t ′ ( + b ) (cid:16) (cid:0) b + b + (cid:1) t ′ + b ( b + ) t ′ + b (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) , (125) and k Large n5 ( t ′ ) c N s = a ( t ′ )= − n q ( t ′ + b ) + t ′ " + t ′ ( t ′ − b − )( t ′ + b ) + t ′ − t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) t ′ + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ (cid:16) t ′ + ( b + ) t ′ − ( b + ) t ′ + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) u (cid:16) t ′ + ( b + ) t ′ ( t ′ − ) + b + (cid:17)(cid:0) ( t ′ + b ) + t ′ (cid:1) + n (cid:0) t ′ + b + (cid:1) + N s t ′ ( + b ) (cid:16)(cid:0) b + b + (cid:1) t ′ + b ( + b ) t ′ + b (cid:17) (cid:0) ( + b ) t ′ + b (cid:1)(cid:0) ( t ′ + b ) + t ′ (cid:1) . (126) D. Comparison of Cumulants obtained from ODEs with those Obtained fromDeterminant Representation
This subsection serves as a check for consistency of our equations. For small values of n wecompute the Hankel determinant from the moments formula (26), since D n ( T , t ) = det (cid:0) m i + j ( T , t ) (cid:1) n − i , j = . The moment generating function in s and t reads M g ( s , t ) = (cid:18) + cs (cid:19) nN s det (cid:16) c i + j ( s , t ) (cid:17) n − i , j = det (cid:16) c i + j ( , t ) (cid:17) n − i , j = (127)where c j ( s , t ) : = t a + j + G ( a + j + ) N s (cid:229) k = (cid:18) N s k (cid:19) ( cs ) k U (cid:18) a + j + , a + j + − k , t + cs (cid:19) , (128)which was derived in Ref. 7. For small fixed integer values of n and N s , e.g., n = , N s = n = , N s =
1, the above determinant can computed without much difficulty, from whichthe cumulants follow. It can be seen that k ( t ) cN s = a ( t ) and k ( t ) c N s = a ( t ) obtained from (127),satisfied third order ODEs for a ( t ) given by (111) and a ( t ) given by (118). Similar results holdfor the higher cumulants, which provide a consistency check. V. LARGE n CORRECTIONS OF CUMULANTS OBTAINED FROM COULOMBFLUID
We have shown that the PDE (104) satisfied by H n ( s , v ) = v ¶ v log M g ( s , v ) k j ( v ) .Under a large n assumption, where v = nt ′ , the first few of these ODEs are approximated as firstorder ODEs for k Large n l ( t ′ ) , whose solutions matched exactly with that obtained from the CoulombFluid analysis for the cumulants k CF l ( t ′ ) .We give a method in this section where the non-linear ODEs generated from the PDE (104) areemployed systematically to obtain “correction terms” to the Coulomb Fluid results. A. Large n expansion of k We assume the first cumulant or k has the following large n expansion a ( t ′ ) = k ( t ′ ) cN s = ne − ( t ′ ) + e ( t ′ ) + ¥ (cid:229) k = e k ( t ′ ) n k . (129)Substituting the above into (113), and setting the coefficients of n j to zero, a system of first orderODEs for e k ( t ′ ) is obtained. These are are solved successively starting from e − ( t ′ ) , followed by e ( t ′ ) and so on.The expression for k Large n1 ( t ′ ) given by (97), gives rise to the initial conditions e − ( ) = e k ( ) = k = , , , , . . . . (130)As a result e − ( t ′ ) is found to satisfy the ODE (cid:18) de − ( t ′ ) dt ′ − n (cid:19) = n ( t ′ + + b ) ( t ′ + b ) + t ′ , (131)and has two solutions e − ( t ′ ) = (cid:18) ∓ b + t ′ ± q ( t ′ + b ) + t ′ (cid:19) , both satisfying the initial condition e ( ) = . We retain the solution that is finite at infinity e − ( t ′ ) = (cid:18) + b + t ′ − q ( t ′ + b ) + t ′ (cid:19) , (132)to match with that obtained from the Coulomb Fluid computation, namely, (97).Continuing, we set the coefficient of n to zero, which implies, de dt ′ = . (133)32etting the coefficient of n to zero gives rise to an ODE involving e ′ − ( t ′ ) , e ′′ − ( t ′ ) , e ′′′ − ( t ′ ) , e ′ ( t ′ ) and e ′ ( t ′ ) . Simplifying, with e − ( t ′ ) given by (132) and e ′ ( t ′ ) given by (133), we find that e ( t ′ ) satisfies: (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) de dt ′ = − t ′ − ( b + ) t ′ + b . (134)Setting the coefficient of the next lowest power of n to zero gives rise to an ODE involving e ′ − ( t ′ ) , e ′′ − ( t ′ ) , e ′′′ − ( t ′ ) , e ′ ( t ′ ) , e ′′ ( t ′ ) , e ′′′ ( t ′ ) , e ′ ( t ′ ) , and e ′ ( t ′ ) . From the expression of e − ( t ′ ) and e ′ ( t ′ ) given by (132) and (133) respectively, we find de dt ′ = . (135)Continue with this process, the next three terms e ( t ′ ) , e ( t ′ ) and e ( t ′ ) are found to satisfy thefollowing equations: (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) de dt ′ = l ( ) ( t ′ ) , (136) de dt ′ = , (137) (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) de dt ′ = l ( ) ( t ′ ) , (138) where l ( ) ( t ′ ) and l ( ) ( t ′ ) are given by l ( ) ( t ′ ) = − t ′ − ( b + ) t ′ + ( b − b − ) t ′ + ( b + )( b − b − ) t ′ − ( b − b − ) b t ′ + ( b + ) b t ′ − b , (139) l ( ) ( t ′ ) = − t ′ + ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ + ( b + )( b − b − ) t ′ − (cid:0) b − b − b + b + (cid:1) t ′ − ( b + )( b − b − b + b + ) t ′ − (cid:0) b + b + b − b − (cid:1) b t ′ + ( b + )( b − b − ) b t ′ + (cid:0) b + b + (cid:1) b t ′ − ( b + ) b t ′ + b . (140) Solving ODEs (133)–(138) with initial conditions e k ( ) =
0, we obtain, k ( t ′ ) cN s = k CF1 ( t ′ ) cN s + ( + b ) t ′ q ( t ′ + b ) + t ′ (cid:229) k = A ( ) k + ( t ′ ) n k + + O (cid:18) n (cid:19) , (141) A ( ) ( t ′ ) , A ( ) ( t ′ ) and A ( ) ( t ′ ) are given by A ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) , (142) A ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) , (143) A ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) . (144) B. Large n Expansion of k In computing a series expansion for the variance, we proceed in a similar way as was just donefor the mean. First, we substitute the expansion for a ( t ′ ) from (141) and a ( t ′ ) = k ( t ′ ) c N s = n f − ( t ′ ) + f ( t ′ ) + ¥ (cid:229) k = f k ( t ′ ) n k , (145)into (118). Equating the coefficients of n k to zero, a system of first order ODEs for f k ( t ′ ) are found.Comparing with the Coulomb Fluid results for k ( t ′ ) in (98) gives rise to the initial conditions f − ( ) = f k ( ) = k = , , , , . . . . (146)In this case, setting the coefficient of n to 0 results in a first order ODE for f − ( t ′ ) : d f − dt ′ = − t ′ + b + (cid:0) ( t ′ + b ) + t ′ (cid:1) / − ( b + ) t ′ (cid:0) ( t ′ + b ) + t ′ (cid:1) / ! . (147)The solution of this, with the initial condition f − ( ) = , reads f − ( t ′ ) = (cid:18) t ′ + + b (cid:19) − q ( t ′ + b ) + t ′ − b + ( b + ) t ′ (cid:16) ( t ′ + b ) + t ′ (cid:17) . (148)Continuing the process, we obtain an ODE for f ( t ′ ) : d f dt ′ = − N s ( b + ) t ′ ( t ′ − b )( t ′ + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) . (149)34he solution with the condition f ( ) = f ( t ′ ) = N s ( + b ) t ′ (cid:0) ( t ′ + b ) + t ′ (cid:1) , (150)from which it can be immediately seen that n f − ( t ′ ) + f ( t ′ ) = k CF2 ( t ′ ) c N s , and we recover k CF2 ( t ′ ) found previously.The ODEs satisfied by f k ( t ′ ) , k = , , , f j ( t ′ ) with appropriate initial conditions.Briefly, the idea is that to determine the ODE satisfied by the k -th correction term f k ( t ′ ) , for k odd, the preceding ( k + ) ODEs satisfied by f j ( t ′ ) , j = − , , . . . , k are employed. For even k ,the previous k ODEs satisfied by f j ( t ′ ) , j = , , . . . , k are employed.Going through the procedure described, we find that k ( t ′ ) has the following large n expansion, k ( t ′ ) c N s = k CF2 ( t ′ ) c N s + ( + b ) t ′ q ( t ′ + b ) + t ′ (cid:229) k = A ( ) k − ( t ′ ) n k − + N s ( + b ) t ′ (cid:229) k = B ( ) k ( t ′ ) n k + O (cid:18) n (cid:19) , (151) where A ( ) ( t ′ ) : = (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) , (152) B ( ) ( t ′ ) : = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) , (153) A ( ) ( t ′ ) : = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) , (154) B ( ) ( t ′ ) : = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) . (155) . Beyond k and k The procedure adopted above for computing the large n expansion series for k ( t ′ ) and k ( t ′ ) easily extends to the higher cumulants. By way of example, here we focus on the third cumulant k . In this case, an asymptotic expansion for a ( t ′ ) is assumed, a ( t ′ ) = k ( t ′ ) c N s = ng − ( t ′ ) + g ( t ′ ) + ¥ (cid:229) k = g k ( t ′ ) n k , (156)along with the initial conditions, g − ( ) = g k ( ) = k = , , , , . . . . (157)In this case, we find that the large n expansion of the third cumulant reads k ( t ′ ) c N s = k CF3 ( t ′ ) c N s + ( + b ) t ′ q ( t ′ + b ) + t ′ (cid:229) k = A ( ) k − ( t ′ ) n k − + N s ( + b ) t ′ (cid:229) k = B ( ) k ( t ′ ) n k + O (cid:18) n (cid:19) , (158) where A ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + N s t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( + b ) (cid:0) ( t ′ + b ) + t ′ (cid:1) ! , (159) B ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) , (160) A ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + N s t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ + t ′ + t ′ b − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) ! , (161) B ( ) ( t ′ ) = (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − b − ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ − t ′ − t ′ b + b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) t ′ ( t ′ + t ′ − t ′ + t ′ b − t ′ b + b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) + t ′ ( t ′ + t ′ − t ′ + t ′ b − t ′ b + b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) − t ′ ( t ′ − t ′ + t ′ b − t ′ b + b + ) (cid:0) ( t ′ + b ) + t ′ (cid:1) . (162) For ease of reference, the ODEs satisfied by g k ( t ′ ) , k = , , , , are placed in Appendix C,equations (C9)–(C12)In summary, this entire section has shown that by expanding the cumulants k ( t ′ ) , k ( t ′ ) and k ( t ′ ) into asymptotic series in n , the Coulomb Fluid results are recovered as the leading ordercontributions in the large n scenario. It is also seen that, by examining the finite- n correction termsfor each cumulant, no terms of O ( n ) or higher are present within the expansions. VI. ASYMPTOTIC PERFORMANCE ANALYSIS BASED ON COULOMB FLUID
In this section, we return to the analysis of the moment generating function, and considerthe high SNR scenario (i.e., as ¯ g → ¥ ). To this end, we will study the Coulomb Fluid basedapproximation derived in Section III (to be complemented in Section VII through analysis basedon Painlev´e equations), where the variables T ′ and t ′ are taken to be dependent on ¯ g , namely, T ′ ( s , ¯ g ) = t ′ ( ¯ g ) + ¯ g sRN s , t ′ ( ¯ g ) = n ( + ¯ g ) N R ˜ b , where ˜ b : = n ¯ g . (163)Note that as ¯ g → ¥ , t ′ ( ¯ g ) −→ N R n n . To obtain the desired high SNR expansion, we compute the moment generating function M g as s → ¥ and ¯ g → ¥ . The Coulomb Fluid based representation (78), when expressed in terms of T ′ ( s , ¯ g ) and N R n n reads M g ( s ) ≈ exp (cid:18) − S AF2 (cid:16) T ′ ( s , ¯ g ) , N R n n (cid:17) − n (cid:20) S AF1 (cid:16) T ′ ( s , ¯ g ) , N R n n (cid:17) − N s log (cid:18) n n T ′ ( s , ¯ g ) N R (cid:19)(cid:21)(cid:19) , (164) where S AF1 and S AF2 are given by (86) and (87) respectively. Our goal is to compute the expansionof M g ( s ) as s → ¥ and ¯ g → ¥ , which turns out to be an expansion in ( ¯ g s ) − . It turns out that thecases b = b = . The Case of b = b =
0, we have N R = n . An easy computation shows that M g admits the followingexpansion: M g ( s ) = ¥ (cid:229) ℓ = A ℓ ( ¯ g s ) d + ℓ/ , (165)where A , A , A , A , . . . are constants independent of s and ¯ g . The leading exponent d is given by d = N s (cid:18) n − N s (cid:19) (166)whilst the first few A ℓ are A = (cid:18) RN s n (cid:19) N s ( n − N s / ) (cid:0) + √ + n (cid:1) nN s N s ( n + N s / ) ( + n ) Ns exp (cid:18) − nN s n (cid:16) − √ + n (cid:17)(cid:19) , (167) r n RN s A = A N s (cid:16) N s √ + n − n (cid:17) , (168) r n RN s A = A (cid:20) N s (cid:16) N s √ + n − n (cid:17) − N s ( + n ) + n ( n − ) (cid:0) N s √ + n − n (cid:1) (cid:21) , (169) r n RN s A = A " N s (cid:16) N s √ + n − n (cid:17) − √ + n + h N s n − n ( N s + )( n − ) − N s ( + n ) i √ + n N s (cid:0) N s √ + n − n (cid:1) − N s ( + n ) − n ( n − ) − n h N s ( + n ) − nN s ( n − ) − n + i N s (cid:0) N s √ + n − n (cid:1) − N s ( + n ) − n ( n − ) . (170) We have refrained from presenting A k , k ≥
4, as these are rather long.
Remark 5.
For the special case ˜ b = ¯ g or n = , implying equal relay and source power, A reducesto the remarkably simple formula:A = ( RN s ) N s ( n − Ns ) j N s n Ns exp (cid:18) N s n j (cid:19) , (171) where j = ( + √ ) / is the Golden ratio.
1. High SNR Analysis of the Symbol Error Rate (SER)
Based on (11), the SER of MPSK modulation can be expanded at high SNR using (165), re-sulting in P MPSK = p ¥ (cid:229) ℓ = A ℓ ( ¯ g g MPSK ) d + ℓ/ I d ,ℓ ( Q ) (172)38here I d ,ℓ ( Q ) = Q Z sin d + ℓ q d q . (173)Considering the first order expansion, following Ref. 48, we may write P MPSK = (cid:16) G a ¯ g (cid:17) − G d + o (cid:16) ¯ g − G d (cid:17) , (174)where we identify G d = N s (cid:18) n − N s (cid:19) (175)as the so-called diversity order , and identify the factor G a = g MPSK (cid:18) A I G d , ( Q ) p (cid:19) − Gd (176)as the so-called array gain (or coding gain ). We note that the result for G d above is consistent witha previous result obtained via a different method in Ref. 8, whilst the expression for G a appearsnew.Whilst it appears that a closed-form solution for the integral (173) is not forthcoming in general(though it can be easily evaluated numerically), such a solution does exist for the important specialcase of BPSK modulation, for which M =
2. In this case we have the particularization, Q = p / I d ,ℓ ( p / ) = √ p G ( d + ℓ/ + / ) G ( d + ℓ/ + ) . (177)Hence, using (176), G a admits the simplified form G a = (cid:18) A √ p G ( G d + / ) G ( G d + ) (cid:19) − Gd . (178)The high SNR results above are illustrated in Fig. 3. The “Simulation” curves are based onnumerically evaluating the exact SER relation (11); the “Coulomb Fluid (Exact)” curves are basedon substituting (164) into (11) and numerically evaluating the resulting integral; the “CoulombFluid (Leading term only)” curves are based on (174); whilst “Coulomb Fluid (Leading 4 terms)”curves are based on the first four terms of (172). The leading-order approximation is shown togive a reasonably good approximation at high SNR, whilst the additional accuracy obtained byincluding a few correction terms is also clearly evident.39 −8 −7 −6 −5 −4 −3 −2 −1 SNR (dB) S E R SimulationCoulomb Fluid (Exact)Coulomb Fluid (Leading term only)Coulomb Fluid (Leading 4 terms) n=3 n=2
FIG. 3. Illustration of the SER versus average received SNR (at relay) ¯ g ; comparison of analysis andsimulations. Results are shown for N R = N D = n and N s =
2, with the full-rate Alamouti OSTBC code (i.e., R = M =
4. The relay power ˜ b is assumed to scale with¯ g by setting ˜ b = ¯ g .
2. High SNR Analysis of the Probability Density Function of g With the moment generating function expansion given above in (165), we may also readilyobtain an approximation for the probability density function (PDF) of g , denoted f g ( x ) , by directLaplace Transform inversion. In particular, we obtain f g ( x ) = ¥ (cid:229) ℓ = A ℓ G ( d + ℓ/ ) x d + ℓ/ − ¯ g d + ℓ/ . (179)For the case of very large ¯ g , with f g ( x ) = A G ( d ) x d − ¯ g d + O (cid:18) g d + / (cid:19) , (180)the leading term gives an approximation for the PDF deep in the left-hand tail. Of course, with theinclusion of more terms, a more refined approximation is obtained.40 . The Case of b = (with N R < N D ) For the situation where b =
0, two sub-cases arise. This first is N R < N D , for which N R = n ;the second is N R > N D , for which N R = ( + b ) n . In the following, we will focus on the first sub-case, N R < N D . It turns out however, that ourresults also apply for N R > N D upon transforming the quantity n to n ∗ by n = n ∗ + b , (181)where n ∗ ( > ) is interpreted as the (fixed) scaling factor between ˜ b and ¯ g , i.e., n ∗ = ˜ b ¯ g . (182)The moment generating function given by (164) admits an expansion distinct from the b = not have fractional powers of 1 / ( ¯ g s ) , reading M g ( s ) = ¥ (cid:229) l = A l ( ¯ g s ) d + l (183)where d = nN s , (184)and A = (cid:16) + b + nb + b q(cid:0) + nb (cid:1) + n (cid:17) Ns ( N s − n b ) (cid:16) + n + nb + q(cid:0) + nb (cid:1) + n (cid:17) nNs ( + b ) n nN s (cid:16)(cid:0) + nb (cid:1) + n (cid:17) Ns (cid:16) + b (cid:17) nNs ( + b ) b Ns ( N s − n b ) × (cid:0) RN s (cid:1) nN s Ns ( n + N s ) exp (cid:18) − nN s n (cid:16) + nb − q(cid:0) + nb (cid:1) + n (cid:17)(cid:19) , (185) A = A N s R nb (cid:20) N s b q ( + nb ) + n − ( n + N s ) b ( + nb ) − N s (cid:21) . (186) In this situation, the sub-leading terms are very complicated, however, the j th term in theexpansion can be written in the following form: A j = A R j N s j + ( j ! ) ( nb ) j (cid:20) E j N s b q ( + nb ) + n + F j (cid:21) , (187)where E j and F j also depend upon n , N s , n and b .41n A , E , and F are given by E = − ( N s + nN s + ) b ( + nb ) − N s − , (188) F = (cid:20) ( N s + n ) (cid:0) N s + nN s + (cid:1) + n ( nN s + ) (cid:21) b ( + nb ) + N s (cid:0) N s + nN s + (cid:1) b ( + nb )+ (cid:0) n − N s + N s (cid:1) b + N s (cid:0) N s + (cid:1) . (189) In A , E , and F are given by E = (cid:20) (cid:0) N s + nN s + n + (cid:1)(cid:0) N s + (cid:1) − n (cid:21) b ( + nb ) + (cid:0) N s + + nN s (cid:1) (cid:0) N s + (cid:1) b ( + nb ) + N s ( − b ) + ( b + ) N s + + nN s b + b (190) F = − (cid:20) (cid:16) ( N s + n ) + N s + n (cid:17) (cid:0) N s + (cid:1) + n (cid:0) − n (cid:1) (cid:21) b ( + nb ) − N s (cid:16) N s + nN s + (cid:17)(cid:16) N s + nN s + (cid:17) b ( + nb ) + (cid:20) ( b − ) N s + ( b − ) nN s − N s − ( + b ) nN s − (cid:0) b n + b + (cid:1) N s − b n (cid:21) b ( + nb ) − ( b + ) (cid:18) n + nN s − N s + N s (cid:19) − N s − N s + n + nN s . (191) In A , E , and F are given by E = − (cid:20)(cid:26) + N s ( N s + n ) + N s (cid:0) + n (cid:1) + nN s (cid:0) + n (cid:1)(cid:27)(cid:0) N s + (cid:1) + nN s (cid:0) − n (cid:1)(cid:21) b ( + nb ) − (cid:20) + N s ( N s + n ) + N s (cid:0) + n (cid:1) + nN s (cid:21)(cid:0) N s + (cid:1) b ( + nb ) + (cid:20) N s ( b − ) + nN s ( b − ) − N s ( b + ) − nN s ( b + ) − N s (cid:0) + b + b n (cid:1) − nN s ( + b ) − ( + b ) (cid:21) b ( + nb ) + N s ( b − ) − N s ( b + ) − nN s b − N s ( b + ) − nN s b − ( + b ) (192) F = (cid:20) n N s + n N s (cid:0) N s + (cid:1) + n N s (cid:0) N s + N s + (cid:1) + n (cid:0) N s + (cid:1)(cid:0) N s + N s + (cid:1) + N s + N s + N s + N s (cid:21) b ( + nb ) + N s (cid:0) N s + nN s + (cid:1)(cid:0) N s + nN s + (cid:1)(cid:0) nN s + + N s (cid:1) b ( + nb ) + " N s b n + (cid:26) N s ( + b ) + N s ( − b ) + N s b (cid:27) n + (cid:26) N s ( − b ) + N s ( − b ) + N s ( + b ) + b (cid:27) n + ( − b ) N s + ( − b ) N s +( + b ) N s + ( b + ) N s b ( + nb ) + (cid:20) (cid:0) + M (cid:1) n N s b + (cid:16) ( − b ) N s + ( b + ) N s + ( b + ) N s + b (cid:17) n ( − b ) N s + ( + b ) N s + ( − b ) N s + ( b + ) N s (cid:21) b ( + nb )+ (cid:0) + b − b (cid:1) N s + (cid:0) − b − b + (cid:1) N s − b n ( b − ) N s + (cid:0) − b + b + (cid:1) N s + (cid:18) + b (cid:19) nN s b + (cid:16) + (cid:0) n + (cid:1) b + b (cid:17) N s + (cid:18) b + (cid:19) n b . (193)
1. High SNR Analysis of the Symbol Error Rate (SER)
Based on (11), the SER of MPSK modulation can be expanded at high SNR using (183) into P MPSK = p ¥ (cid:229) ℓ = A ℓ ( ¯ g g MPSK ) d + ℓ I d + ℓ ( Q ) (194)where I r ( Q ) = Z Q sin r q d q . (195)Note that here, in contrast to (173), the exponent r is a positive integer . As such, (195) admits thefollowing closed-form solution: (Ref. 49, 2.513.1) I r ( Q ) = Q r (cid:18) rr (cid:19) + ( − ) r r − r − (cid:229) j = ( − ) j (cid:18) rj (cid:19) sin (cid:16) ( r − j ) Q (cid:17) ( r − j ) . (196)As before, a first-order approximation is of key interest, giving P MPSK = (cid:16) G a ¯ g (cid:17) − G d + o (cid:16) ¯ g − G d (cid:17) , (197)where we identify the diversity order G d = nN s , (198)and the array gain G a = g MPSK (cid:18) A I G d ( Q ) p (cid:19) − Gd . (199)The result for G d above is consistent with a result obtained via a different method in Ref. 8, whilstthe expression for G a appears new. The high SNR results above, for the case b =
0, are illustratedin Fig. 4. As before, the “Simulation” curves are based on numerically evaluating the exact SERrelation (11), and the “Coulomb Fluid (Exact)” curves are based on substituting (164) into (11) andnumerically evaluating the resulting integral. Moreover, the “Coulomb Fluid (Leading term only)”43urves are based on (197), whilst the “Coulomb Fluid (Leading 5 terms)” curves are based on thefirst five terms of (194). Again, the leading-order approximation is shown to give a reasonablygood approximation at high SNR, whilst the additional accuracy obtained by including a fewcorrection terms is also evident. −8 −6 −4 −2 SNR (dB) S E R SimulationCoulomb Fluid (Exact)Coulomb Fluid (Leading term only)Coulomb Fluid (Leading 5 terms)N D =6 N D =3 FIG. 4. Illustration of the SER versus average received SNR (at relay) ¯ g ; comparison of analysis andsimulations. Results are shown for N R = N s =
2, with the full-rate Alamouti OSTBC code (i.e., R = M =
4. The relay power ˜ b is assumed to scale with¯ g by setting ˜ b = ¯ g .
2. High SNR Analysis of the Probability Density Function of g As before, based on the moment generating function expansion (183), applying for b =
0, wecan immediately take a Laplace inversion to obtain the following high SNR representation for thePDF of g , f g ( x ) = ¥ (cid:229) ℓ = A ℓ G ( d + ℓ ) x d + ℓ − ¯ g d + ℓ . (200)44e are mainly interested in the leading order term, and so we write the above as f g ( x ) = A G ( d ) x d − ¯ g d + O (cid:18) g d + (cid:19) . (201)Note that despite the similarity with (180), interestingly, these results do not coincide upon tak-ing b → d and A . This seems to indicate that the doubleasymptotics ¯ g → ¥ and b → VII. CHARACTERIZING A THROUGH PAINLEV ´E V In this section, we obtain the leading term of the large s and large ¯ g expansion (183) from aPainlev´e V differential equation, thus demonstrating the accuracy of the Coulomb Fluid approxi-mation. We will focus in this section on the case b = g , regarded a function of s and t , given by(21), reads, M g ( s , t ) = K n , a (cid:18) + cs (cid:19) nN s n ! Z [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x a k e − x k t + x kt + cs + x k ! N s dx k , (202)where K n , a is a normalization constant in (27). We also recall that c and t are given by c = ¯ g RN s , t = ( + ¯ g ) N R ˜ b , where ˜ b : = n ¯ g , (203)respectively. So as ¯ g → ¥ , we note that t −→ N R n . A simple computation shows that M g ( s , t ) admits the following expansion for large ¯ g s : M g ( s , t ) = ( RN s ) nN s K n , a ( ¯ g s ) nN s (cid:18) − nN s ¯ g s + . . . (cid:19) × n ! Z [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:213) k = x a − N s k e − x k ( t + x k ) N s (cid:18) − tRN s ¯ g sx k + . . . (cid:19) dx k = ( RN s ) nN s D n [ w dLag ( · , , a − N s , N s )] ( ¯ g s ) nN s D n (cid:2) w dLag ( · , t , a − N s , N s ) (cid:3) − R nN s + N nN s + s D n [ w dLag ( · , , a − N s , N s )] ( ¯ g s ) nN s + " nD n (cid:2) w dLag ( · , t , a − N s , N s ) (cid:3) + tn ! Z [ , ¥ ) n (cid:213) ≤ i < j ≤ n ( x j − x i ) n (cid:229) l = x − l ! n (cid:213) k = x a − N s k e − x k ( t + x k ) N s dx k + O (cid:18) ( ¯ g s ) nN s + (cid:19) , (204) w dLag ( x , t , a − N s , N s ) is the deformation of the classical Laguerre weight, i.e., w dLag ( x , t , a − N s , N s ) = x a − N s e − x ( t + x ) N s , t > , a − N s > − , (205)and K n , a = D n [ w dLag ( · , , a − N s , N s )] is independent of t .The condition a − N s > − w ( a − N s ) Lag ( x ) (see Ref. 9). This, in turn, will ensure the validityof the first two terms in (204), while for the multiple integral in the third term to converge, thecondition a − N s > a and N s are integers, satisfying a ≥ N s >
0. Therefore a ≥ N s implies that the result for A presented below is valid for a > d = nN s , (206)and A = ( RN s ) nN s D n [ w dLag ( · , t , a − N s , N s )] D n [ w dLag ( · , , a − N s , N s )] . (207)We see that A is up to a constant the Hankel determinant which generates a particular Painlev´e Vand shows up in the single-user MIMO problem studied in Ref. 13.We obtain A , through a large n expansion of D n [ w dLag ( · , t , a − N s , N s )] D n [ w dLag ( · , , a − N s , N s )] , for a > . We will see that A precisely matches that obtained in (185).From Ref. 13 we learned that the logarithmic derivative of D n [ w dLag ( · , t , a − N s , N s )] with re-spect to t , H n ( t ) : = t ddt log D n [ w dLag ( · , t , a − N s , N s )]= t ddt log (cid:18) D n [ w dLag ( · , t , a − N s , N s )] D n [ w dLag ( · , , a − N s , N s )] (cid:19) (208)satisfies the Painlev´e V: (cid:16) tH ′′ n (cid:17) = h(cid:0) t + n + a (cid:1) H ′ n − H n + nN s i − (cid:16) tH ′ n − H n + n ( n + a ) (cid:17)(cid:16) H ′ n (cid:17)(cid:16) H ′ n + N s (cid:17) , (209)where ′ denote derivative w.r.t. t . 46e restrict to the case where N R < N D , for which N R = n . The case where N R > N D can beconsidered in a similar fashion as outlined in Section VI B. Setting a = n b in the above equation,where b = mn − t = n n , (210)an easy computation shows that Y n ( n ) : = H n ( n / n ) satisfies n n (cid:16) Y ′ n + n Y ′′ n (cid:17) = (cid:20) n (cid:16) ( b + ) n + (cid:17) Y ′ n + Y n − nN s (cid:21) + n n (cid:16) − n Y ′ n − Y n + ( + b ) n (cid:17)(cid:16) Y ′ n (cid:17)(cid:16) − n Y ′ n + nN s (cid:17) , (211)where ′ denotes derivative with respect to n .We seek a solution for Y n ( n ) in the form Y n ( n ) = np − ( n ) + p ( n ) + ¥ (cid:229) j = p j ( n ) n j , (212)from which A ( n ) is found to be A ( n ) = ( RN s ) nN s exp − n Z ¥ np − ( n ′ ) + p ( n ′ ) + (cid:229) ¥ j = n − j p j ( n ′ ) n ′ d n ′ , (213) ≈ ( RN s ) nN s exp − n Z ¥ np − ( n ′ ) + p ( n ′ ) n ′ d n ′ " − n n Z ¥ p ( n ′ ) n ′ d n ′ + . . . . (214) Substituting (212) into (211) leads to (211) taking the form c − n + c − n + c + ¥ (cid:229) j = c j n − j = , (215)where c − depends on p − ( n ) and its derivatives, and c i , i = − , , , , . . . depend on p − ( n ) , p ( n ) up to p i + ( n ) and their derivatives. Of course, each c i also depends upon n , N s and b .Assuming that the coefficient of n k is zero, we find that the equation c − = (cid:20) n (cid:16) ( b + ) n + (cid:17) p ′− ( n ) + p − ( n ) − N s (cid:21) = n ( + b ) p ′− ( n ) (cid:16) n p ′− ( n ) + N s (cid:17) . (216)47ith MAPLE, the solutions of the above ODE for p − ( n ) are found to be p − ( n ) = (cid:16) − nb − + q(cid:0) + nb (cid:1) + n (cid:17) N s n , (217) p − ( n ) = (cid:16) − nb − − q(cid:0) + nb (cid:1) + n (cid:17) N s n , (218) p − ( n ) = (cid:18) + b + n (cid:19) C + N s + p ( + b ) C ( N s + C ) , (219)where C is a constant of integration. .The equation c − = p − ( n ) and p ( n ) givenby (cid:20) n (cid:16) p ′− ( n ) (cid:17) + n p ′− ( n ) p − ( n ) + n ( − nb ) p ′ ( n ) + p ( n ) (cid:21) (cid:16) N s − n p − ( n ) (cid:17) − n (cid:20)(cid:16) b n + ( b + ) n + n (cid:17) p ′ ( n ) − n p − ( n ) + (cid:16) + ( b + ) n (cid:17) p ( n ) (cid:21) p ′− ( n ) − n (cid:16) p ′− ( n ) (cid:17) = (cid:16) n ( n + ) ( bn + ) p ′ ( n ) − ( n − ) p ( n ) (cid:17) p − ( n ) . (220) With p − ( n ) given by (217), chosen to match the result from the Coulomb Fluid (185), we findthat the first order ODE in p ( n ) has the solution, p ( n ) = (cid:16) + b + nb − b q(cid:0) + nb (cid:1) + n (cid:17) n N s (cid:16)(cid:0) + nb (cid:1) + n (cid:17) . (221)We disregard the second and third solutions for p − ( n ) ; as these would lead to p ( n ) which donot generate the A in agreement with that obtained from the Coulomb Fluid method.Substituting p − ( n ) from (217) and p ( n ) from (221) into c = p ( n ) = N s n h(cid:16) b ( + b ) n + (cid:0) b + b + (cid:1) n + + b (cid:17) N s − ( + b ) n i(cid:16)(cid:0) + nb (cid:1) + n (cid:17) / − N s n ( + ( + b ) n ) b (cid:16)(cid:0) + nb (cid:1) + n (cid:17) . (222) Hence, A ( n ) has a large n expansion, A ( n ) = q ( n ) " + q ( n ) n + O (cid:18) n (cid:19) , (223) Note that the constants of integration in (217) and (218) are zero. q ( n ) = (cid:16) + b + nb + b q(cid:0) + nb (cid:1) + n (cid:17) Ns ( N s − n b ) (cid:16) + n + nb + q(cid:0) + nb (cid:1) + n (cid:17) nNs ( + b ) n nN s (cid:16)(cid:0) + nb (cid:1) + n (cid:17) Ns (cid:16) + b (cid:17) nNs ( + b ) b Ns ( N s − n b ) × (cid:0) RN s (cid:1) nN s Ns ( n + N s ) exp (cid:18) − nN s n (cid:16) + nb − q(cid:0) + nb (cid:1) + n (cid:17)(cid:19) . (224) The leading term of A in (223) agrees precisely with the A computed via the Coulomb Fluidmethod in (185).Using a method similar to the cumulant analysis of Section V, we can also compute the firstcorrection term to A (i.e., the quantity q ( n ) ) using (222), which reads q ( n ) = N s (cid:16) (cid:0) N s − (cid:1) b + (cid:0) N s − (cid:1) ( + b ) (cid:17) ( + b ) " b − n ( b n + b + ) (cid:16)(cid:0) + nb (cid:1) + n (cid:17) / − ( bn + n + ) N s b (cid:16)(cid:0) + nb (cid:1) + n (cid:17) − ( + b ) (cid:0) N s − (cid:1) N s (cid:16)(cid:0) + nb (cid:1) + n (cid:17) / ( + b ) " ( + b ) n + + ( + b ) N s n (cid:16)(cid:0) + nb (cid:1) + n (cid:17) / . (225) Higher order corrections could also be obtained in a similar way.
VIII. CONCLUSION
In this paper, we have introduced two methods for characterizing the received SNR distributionin a certain MIMO communication system adopting AF relaying. We showed that the mathe-matical problem of interest pertains to computing a certain Hankel determinant generated by aparticular two-time deformation of the classical Laguerre weight. By employing the ladder oper-ator approach, together with Toda-type evolution equations in the time variables, we establishedan exact representation of the Hankel determinant in terms of a double-time PDE, which reducesto a Painlev´e V in various limits. This result yields an exact and fundamental characterization ofthe SNR distribution, through its moment generating function. Complementary to the exact rep-resentation, we also introduced the linear statistics Coulomb Fluid approach as an efficient wayto compute very quickly the asymptotic properties of the moment generating function for suffi-ciently large dimensions (i.e., for sufficiently large numbers of antennas). These results–which49ave only started to be used recently in problems related to wireless communications and infor-mation theory–produced simple closed-form approximations for the moment generating function.These were employed to yield simple closed-form approximations for the error probability (for aclass of M -PSK digital modulation), which were shown via simulations to be remarkably accurate,even for very small dimensions.To further demonstrate the utility of our methodology, we employed our asymptotic CoulombFluid characterization in conjunction with the PDE representation to provide a rigorous studyof the cumulants of the SNR distribution. Starting with a large- n framework, we computed inclosed-form the finite- n corrections to the first few cumulants. It was seen that the Coulomb Fluidapproach supplies the crucial initial conditions which are instrumental in obtaining asymptoticexpansions from the PDE. We also derived asymptotic properties of the moment generating func-tion when the average SNR was sufficiently high, and in such regime extracted key performancequantities of engineering interest, namely, the array gain and diversity order. ACKNOWLEDGMENTS
N. S. Haq is supported by an EPSRC grant. M. R. McKay is supported by the Hong KongResearch Grants Council under Grant No. 616911.50 ppendix A: Characterization of Hankel Determinant Using the Theory of OrthogonalPolynomials
In this section, we describe the process by which we characterize the Hankel determinant byusing the theory of orthogonal polynomials and their associated ladder operators. See Refs. 9, 13,14, 16, and 19 for the background to this theory.Consider a sequence of polynomials { P n ( x ) } orthogonal with respect to the weight function w AF ( x , T , t ) , given by (22) w AF ( x , T , t ) = x a e − x (cid:18) t + xT + x (cid:19) N s , ≤ x < ¥ , (A1)i.e., ¥ Z P n ( x ) P m ( x ) w AF ( x , T , t ) dx = h n d n , m , (A2)where h n is the square of the L norm of P n ( x ) . The Hankel determinant (1) is reduced to thefollowing product: D n [ w AF ] = n − (cid:213) j = h j . (A3)Our convention is to write P n ( x ) as P n ( x ) : = x n + p ( n ) x n − + p ( n ) x n − + · · · + P n ( ) . (A4)Hence, this implies that properties of the Hankel determinant may be obtained by characterizingthe class of polynomials which are orthogonal with respect to w AF ( x , T , t ) , over [ , ¥ ) . It is clearthat the coefficients of the polynomial P n ( x ) , p i ( n ) , will depend on T , t , a and N s ; for brevity, wedo not display this dependence.From the orthogonality relation, the three term recurrence relation follows: xP n ( x ) = P n + ( x ) + a n P n ( x ) + b n P n − ( x ) , n = , , , . . . (A5)with initial conditions P ( x ) ≡ , and b P − ( x ) ≡ . The main aim is to determine these unknown recurrence coefficients a n and b n from the givenweight. Substituting (A4) into the three term recurrence relation results in a n = p ( n ) − p ( n + ) (A6)51here p ( ) : =
0. Taking a telescopic sum gives n − (cid:229) j = a j = − p ( n ) . (A7)Moreover, combining the orthogonality relationship (A2) with the three term recurrence relationleads to b n = h n h n − , (A8)which can also be expressed in terms of the Hankel determinant D n in (A3) through b n = D n + D n − D n , (A9)since h n = D n + D n . Ladder Operators, Compatibility Conditions, and Difference Equations
In the theory of Hermitian random matrices, orthogonal polynomials plays an important role,since the fundamental object, namely, Hankel determinants or partition functions, are expressedin terms of the associated L norm, as indicated for example in (A3). Moreover, as indicatedabove, the Hankel determinants are intimately related to the recurrence coefficients a n and b n ofthe orthogonal polynomials (for other recent examples, see Refs. 19, 50–52).As we now show, there is a recursive algorithm that facilitates the determination of the recur-rence coefficients a n and b n . This is implemented through the use of so-called “ladder operators”as well as their associated compatibility conditions. This approach can be traced back to Laguerreand Ref. 53. Recently, Magnus applied ladder operators to non-classical orthogonal polynomialsassociated with random matrix theory and the derivation of Painlev´e equations, while Ref. 10 usedthe associated compatibility conditions in the study of finite n matrix models. See Refs. 14, 16, and19 for other examples of the application of this approach.From the weight function w AF ( x , T , t ) , one constructs the associated potential v ( x ) through v ( x ) = − log w AF ( x ) = x − a log x − N s log (cid:18) t + xT + x (cid:19) , (A10)and therefore, v ′ ( x ) = − a x − N s x + t + N s x + T . (A11)52s shown in Ref. 15, a pair of ladder operators, to be satisfied by our orthogonal polynomials ofinterest, are expressed in terms of v ( x ) and are given by (cid:20) ddx + B n ( x ) (cid:21) P n ( x ) = b n A n ( x ) P n − ( x ) , (cid:20) ddx − B n ( x ) − v ′ ( x ) (cid:21) P n − ( x ) = − A n − ( x ) P n ( x ) , (A12)where A n ( x ) = h n ¥ Z v ′ ( x ) − v ′ ( y ) x − y P n ( y ) w AF ( y ) dy , B n ( x ) = h n − ¥ Z v ′ ( x ) − v ′ ( y ) x − y P n ( y ) P n − ( y ) w AF ( y ) dy , (A13)and where, for the sake of brevity, we have dropped the t , T dependence in w AF . Moreover, there are associated fundamental compatibility conditions to be satisfied by A n ( x ) and B n ( x ) , which are given by B n + ( x ) + B n ( x ) = ( x − a n ) A n ( x ) − v ′ ( x ) , ( S )1 + ( x − a n )[ B n + ( x ) − B n ( x )] = b n + A n + ( x ) − b n A n − ( x ) . ( S )These were initially derived for any polynomial v ( x ) (see Refs. 54–56), and then were shown tohold for all x ∈ C ∪ { ¥ } in greater generality. We now combine ( S ) and ( S ) as follows. First, multiplying ( S ) by A n ( x ) , it can be seen thatthe RHS is a first order difference, while ( x − a n ) A n ( x ) on the LHS can be replaced by B n + ( x ) + B n ( x ) + v ′ ( x ) from ( S ) . Then, taking a telescopic sum with initial conditions B ( x ) = A − ( x ) = n − (cid:229) j = A j ( x ) + B n ( x ) + v ′ ( x ) B n ( x ) = b n A n ( x ) A n − ( x ) . ( S ′ )The condition ( S ′ ) is of considerable interest, since it is intimately related to the logarithm of theHankel determinant. In order to gain further information about the determinant, we need to find away to reduce the sum to fixed number of quantities; for which, ( S ′ ) ultimately provides a way ofgoing forward. 53 emark 6. Since our v ′ ( x ) is a rational function of x, we see that v ′ ( x ) − v ′ ( y ) x − y = a xy + N s ( x + t )( y + t ) − N s ( x + T )( y + T ) , (A14) is also a rational function of x, which in turn implies that A n ( x ) and B n ( x ) are rational functionsof x. Consequently, equating the residues of the simple and double pole at x = , x = − T , x = − ton both sides of the compatibility conditions ( S ) , ( S ) and ( S ′ ) , we obtain equations containingnumerous n, T and t dependant quantities; which we call the “auxiliary variables” (to be intro-duced below). The resulting non-linear discrete equations are likely very complicated, but themain idea is to express the recurrence coefficients a n and b n in terms of these auxiliary variables,and eventually take advantage of the product representation (A3) to obtain an equation satisfiedby the logarithmic derivative of the Hankel determinant. Now substituting (A14) into (A13) which define A n ( x ) and B n ( x ) , and followed by integrationby parts, we find A n ( x ) = R n ( T , t ) + − R ∗ n ( T , t ) x + R ∗ n ( T , t ) x + t − R n ( T , t ) x + T , (A15) B n ( x ) = r n ( T , t ) − n − r ∗ n ( T , t ) x + r ∗ n ( T , t ) x + t − r n ( T , t ) x + T , (A16)where R ∗ n ( T , t ) ≡ N s h n ¥ Z w AF ( y ) P n ( y ) y + t dy , r ∗ n ( T , t ) ≡ N s h n − ¥ Z w AF ( y ) P n ( y ) P n − ( y ) y + t dy , R n ( T , t ) ≡ N s h n ¥ Z w AF ( y ) P n ( y ) y + T dy , r n ( T , t ) ≡ N s h n − ¥ Z w AF ( y ) P n ( y ) P n − ( y ) y + T dy , (A17) are the auxiliary variables. Difference Equations from Compatibility Conditions
Inserting A n ( x ) and B n ( x ) into the compatibility conditions ( S ) , ( S ) and ( S ′ ) , and equatingthe residues as described above yields a system of 12 equations. Note that there are actually 14equations, the compatibility conditions ( S ) and ( S ) also have O ( x ) terms which yield equationsthat lead to 0 = ( S ) give the following set of equations, r n + + r n − r ∗ n + − r ∗ n − n − = a − a n ( R n + − R ∗ n ) , (A18) r ∗ n + + r ∗ n = N s − ( a n + t ) R ∗ n , (A19) − r n + − r n = − N s + ( a n + T ) R n . (A20)54imilarly, the compatibility condition ( S ) gives the following set of equations, − a n ( r n + − r n − − r ∗ n + + r ∗ n ) = b n + − b n + b n + ( R n + − R ∗ n + ) − b n ( R n − − R ∗ n − ) , (A21) ( t + a n )( r ∗ n − r ∗ n + ) = b n + R ∗ n + − b n R ∗ n − , (A22) ( T + a n )( r n − r n + ) = b n + R n + − b n R n − . (A23)To obtain a system of difference equations from the compatibility condition ( S ′ ) is somewhatmore complicated, but carrying out the process, equating all respective residues in ( S ′ ) yields sixequations. The first three are obtained by equating the residues of the double pole at x = x = − t and x = − T respectively: − r n r ∗ n − ( n − N s + a ) r n +( n + N s + a ) r ∗ n + n ( n + a ) = b n − b n ( R n R ∗ n − + R ∗ n R n − )+ b n ( R n + R n − ) − b n ( R ∗ n + R ∗ n − ) , (A24) r ∗ n ( r ∗ n − N s ) = b n R ∗ n R ∗ n − , (A25) r n ( r n − N s ) = b n R n R n − , (A26)while the last three are given by equating the residues of the simple pole at x = x = − t and x = − T respectively: n − (cid:229) j = ( R j − R ∗ j ) + ( r ∗ n − N s )( r n − n − r ∗ n ) − a r ∗ n t + ( N s − r n )( r n − n − r ∗ n ) + a r n T + r n − r ∗ n = (cid:18) T + t (cid:19) b n ( R ∗ n R n − + R n R ∗ n − ) + b n ( R ∗ n + R ∗ n − − R ∗ n R ∗ n − ) t − b n ( R n + R n − + R n R n − ) T , (A27) n − (cid:229) j = R ∗ j − ( r ∗ n − N s )( r n − n − r ∗ n ) − a r ∗ n t + ( N s − r n ) r ∗ n + N s r n T − t + r ∗ n = − b n ( R ∗ n + R ∗ n − − R ∗ n R ∗ n − ) t − (cid:18) t + T − t (cid:19) b n ( R ∗ n R n − + R n R ∗ n − ) , (A28) n − (cid:229) j = R j + ( N s − r n )( r n − n − r ∗ n ) + a r n T + ( N s − r n ) r ∗ n + N s r n T − t + r n = − b n ( R n + R n − + R n R n − ) T + (cid:18) T − T − t (cid:19) b n ( R ∗ n R n − + R n R ∗ n − ) . (A29)55 emark 7. It can be seen that (A27) is a combination of (A28) and (A29), and serves as a consis-tency check. We also see that (A28) and (A29) respectively gives us the value of (cid:229) j R ∗ j and (cid:229) j R j automatically in closed form. Hence we can also obtain any linear combination of these sumsin closed form, which is a crucial step in obtaining a link between a linear combination of thelogarithmic partial derivatives of the Hankel determinant, and the quantities b n , r n and r ∗ n . Thisstep would not be possible without ( S ′ ) , also known as the bilinear identity . Analysis of Non-linear System
Whilst some of the above difference equations (A18)-(A29) look rather complicated, our aimis to manipulate these equations in such a way as to give us insight into the recurrence coefficients a n and b n . Our dominant strategy is to always try to describe the recurrence coefficients a n and b n in terms of the auxiliary variables R n , r n , R ∗ n and r ∗ n .The first thing we can do is to sum (A18) through (A20) to get a simple expression for therecurrence coefficient a n in terms of the auxiliary variables R n and R ∗ n : a n = T R n − tR ∗ n + a + n + . (A30)We can immediately see that by taking a telescopic sum of the above from j = j = n −
1, andrecalling equation (A7), gives rise to T n − (cid:229) j = R j − t n − (cid:229) j = R ∗ j + n ( n + a ) = n − (cid:229) j = a j , = − p ( n ) . (A31)Looking at ( S ) next, we sum equations (A21) through (A23) to get a n = b n + − b n + T ( r n − r n + ) + t ( r ∗ n + − r ∗ n ) . (A32)Taking a telescopic sum of the above from j = j = n − b n = Tr n − tr ∗ n − p ( n ) . (A33)We are now in a position to derive the following important lemma, which describes the recurrencecoefficients a n and b n in terms of the set of auxiliary variables:56 emma 2. The quantities a n and b n are given in terms of the auxiliary variables R n , r n , R ∗ n andr ∗ n as a n = T R n − tR ∗ n + a + n + , (A34) b n = + R n − R ∗ n (cid:20) ( + R n ) r ∗ n ( r ∗ n − N s ) R ∗ n + ( R ∗ n − ) r n ( r n − N s ) R n − r n r ∗ n − ( n − N s + a ) r n + ( n + N s + a ) r ∗ n + n ( n + a ) (cid:21) . (A35) Proof.
Equation (A34) is a restatement of equation (A30).To obtain (A35), we use the equations obtained from ( S ′ ) . We eliminate R ∗ n − and R n − from(A24) using (A25) and (A26) respectively, and then rearrange to express b n in terms of the auxil-iary quantities. Toda Evolution
In this section, n is kept fixed while we vary two parameters in the weight function (22), namely T and t . The other parameters, a and N s are kept fixed.Differentiating (A2), w.r.t. T and t , for m = n , gives ¶ T ( log h n ) = − R n , (A36) ¶ t ( log h n ) = R ∗ n (A37)respectively. Then, from equation (A8), i.e. b n = h n / h n − , it follows that ¶ T b n = b n ( R n − − R n ) , (A38) ¶ t b n = b n ( R ∗ n − R ∗ n − ) . (A39)Applying ¶ T and ¶ t to the orthogonality relation ¥ Z w AF ( x , T , t ) P n ( x ) P n − ( x ) dx = ¶ T p ( n ) = r n , (A40) ¶ t p ( n ) = − r ∗ n . (A41)57ote that in the above computations with ¶ t and ¶ T we must keep in mind that the coefficientsof P n ( x ) depend on t and T . Using the above two equations, we get that ( T ¶ T + t ¶ t ) p ( n ) = Tr n − tr ∗ n , = p ( n ) + b n , (A42)where the second equality follows from (A33).Using equation (A6), i.e. a n = p ( n ) − p ( n + ) , we get that ¶ T a n = r n − r n + , (A43) ¶ t a n = r ∗ n + − r ∗ n . (A44)Now, combining (A43), (A44), (A33), and (A6), and similarly using (A34), (A38) and (A39),we arrive at the following lemma: Lemma 3.
The recurrence coefficients a n and b n satisfy the following partial differential relations: ( T ¶ T + t ¶ t − ) a n = b n − b n + , ( T ¶ T + t ¶ t − ) b n = b n ( a n − − a n ) . In the second of the above two equations, we eliminate a n using the first equation to derive asecond order differential-difference relation for b n : ( T ¶ T T + T t ¶ Tt + t ¶ tt ) log b n = b n − − b n + b n + − , which is a two-variable generalization of the Toda molecule equations. Now, before proceeding to examine the time-evolution behaviour of the Hankel determinant,we state the following lemmas regarding the auxiliary variables R n , R ∗ n , r n and r ∗ n . Lemma 4.
The auxiliary variables R n , R ∗ n , r n and r ∗ n satisfy the following first order PDE system:T ¶ T R n − t ¶ T R ∗ n = h T R n − tR ∗ n + n + a + T i R n + r n − N s , (A45) T ¶ t R n − t ¶ t R ∗ n = h − T R n + tR ∗ n − n − a − t i R ∗ n − r ∗ n + N s , (A46)58 ¶ T r n − t ¶ T r ∗ n = r n ( r n − N s ) " R n − R ∗ n − + R n − R ∗ n − R n ( + R n ) R ∗ n ( + R n − R ∗ n ) r ∗ n ( r ∗ n − N s )+ R n + R n − R ∗ n h r n r ∗ n + ( n − N s + a ) r n − ( n + N s + a ) r ∗ n − n ( n + a ) i , (A47) T ¶ t r n − t ¶ t r ∗ n = r ∗ n ( r ∗ n − N s ) " − R ∗ n + + R n + R n − R ∗ n + R ∗ n ( R ∗ n − ) R n ( + R n − R ∗ n ) r n ( r n − N s ) − R ∗ n + R n − R ∗ n h r n r ∗ n + ( n − N s + a ) r n − ( n + N s + a ) r ∗ n − n ( n + a ) i . (A48) Proof.
To obtain (A45), we substitute in for a n in (A43) using (A34) to yield r n − r n + = ¶ T a n , = R n + T ¶ T R n − t ¶ T R ∗ n . (A49)Eliminating r n + in the above formula using (A20) gives2 r n = h − a n − T i R n + T ¶ T R n − t ¶ T R ∗ n + N s . (A50)Finally, we eliminate a n again using (A34), and then rearrange to obtain (A45).We obtain (A46) in a similar method to (A45). This time, we first substitute in for a n in (A44)using (A34). We then proceed to eliminate r ∗ n + using (A19), and finally eliminate a n again using(A34) to obtain (A46).To obtain (A47), we differentiate equation (A33) with respect to T . Following this, we substi-tute in for ¶ T b n and ¶ T p ( n ) using (A38) and (A40) respectively to yield T ¶ T r n − t ¶ T r ∗ n = b n ( R n − − R n ) . (A51)We then proceed to replace b n R n − by r n ( r n − N s ) / R n using (A26), and finally we eliminate b n infavor of R n , R ∗ n , r n and r ∗ n using (A35) to obtain equation (A47).Similarly, we obtain (A48) by first differentiating (A33) with respect to t . Following this, wesubstitute in for ¶ t b n and ¶ t p ( n ) using (A39) and (A41) respectively to yield T ¶ t r n − t ¶ t r ∗ n = b n ( R ∗ n − R ∗ n − ) . (A52)We then proceed to replace b n R ∗ n − by r ∗ n ( r ∗ n − N s ) / R ∗ n using (A25), and finally we eliminate b n infavor of R n , R ∗ n , r n and r ∗ n using (A35) to obtain equation (A48), completing our proof.From Lemma 4, the auxiliary variables r n and r ∗ n appear linearly in equations (A45) and (A46)respectively. By eliminating r n and r ∗ n from (A47) and (A48) using (A45) and (A46), we obtain59 second order PDE system for the auxiliary variables R n and R ∗ n . This is stated in the followinglemma: Lemma 5.
The auxiliary variables R n and R ∗ n satisfy the following second-order PDE system: = T (cid:16) T ¶ TT + t ¶ Tt (cid:17) R n − t (cid:16) T ¶ TT + t ¶ tt (cid:17) R ∗ n − (cid:16) ( + R n )( − R ∗ n ) + R n (cid:17) R n ( + R n − R ∗ n ) (cid:20) T (cid:16) ¶ T R n (cid:17) − t (cid:16) ¶ T R ∗ n (cid:17)(cid:21) + R n ( + R n ) R ∗ n ( + R n − R ∗ n ) (cid:20) T (cid:16) ¶ t R n (cid:17) − t (cid:16) ¶ t R ∗ n (cid:17)(cid:21) + R n + R n − R ∗ n (cid:20) T (cid:16) ¶ T R n (cid:17) − t (cid:16) ¶ T R ∗ n (cid:17)(cid:21)(cid:20) T (cid:16) ¶ t R n (cid:17) − t (cid:16) ¶ t R ∗ n (cid:17)(cid:21) + T ( tR ∗ n + ) (cid:16) ¶ T R n (cid:17) + t ( T R n + ) (cid:16) ¶ t R n (cid:17) − t (cid:20) R ∗ n (cid:16) ¶ T R ∗ n (cid:17) + R n (cid:16) ¶ t R ∗ n (cid:17)(cid:21) − t ( T − t ) (cid:16) ¶ T R ∗ n (cid:17) − R n ( T R n − tR ∗ n + T ) (cid:18) T R n − tR ∗ n + n + a + + t (cid:19) − T ( T − t ) R n ( R n + ) − (cid:16) R n ( R n − R ∗ n ) − R ∗ n (cid:17) N s R ∗ n R n − a R n + R n − R ∗ n , (A53)0 = T (cid:16) T ¶ Tt + t ¶ tt (cid:17) R n − t ( T + t ) (cid:16) ¶ tt R ∗ n (cid:17) + R ∗ n ( − R ∗ n ) R n ( + R n − R ∗ n ) (cid:20) T (cid:16) ¶ T R n (cid:17) − t (cid:16) ¶ T R ∗ n (cid:17)(cid:21) + (cid:16) ( + R n )( − R ∗ n ) − R ∗ n (cid:17) R ∗ n ( + R n − R ∗ n ) (cid:20) T (cid:16) ¶ t R n (cid:17) − t (cid:16) ¶ t R ∗ n (cid:17)(cid:21) − R ∗ n + R n − R ∗ n (cid:20) T (cid:16) ¶ T R n (cid:17) − t (cid:16) ¶ T R ∗ n (cid:17)(cid:21)(cid:20) T (cid:16) ¶ t R n (cid:17) − t (cid:16) ¶ t R ∗ n (cid:17)(cid:21) + T ( tR ∗ n + ) (cid:16) ¶ T R ∗ n ) + t ( T R n + ) (cid:16) ¶ t R ∗ n (cid:17) − T (cid:20) R ∗ n (cid:16) ¶ T R n (cid:17) + R n (cid:16) ¶ t R n (cid:17)(cid:21) − T ( T − t ) (cid:16) ¶ t R n (cid:17) + R ∗ n ( T R n − tR ∗ n + t ) (cid:18) T R n − tR ∗ n + n + a + + T (cid:19) − t ( T − t ) R ∗ n ( − R ∗ n )+ (cid:16) R ∗ n ( R n − R ∗ n ) − R n (cid:17) N s R ∗ n R n + a R ∗ n + R n − R ∗ n . (A54) Toda Evolution of Hankel Determinant
Recall that the moment generating function is related to the Hankel determinant through equa-tion (24): M g ( T , t ) = D n [ w ( a ) Lag ( · )] (cid:18) Tt (cid:19) nN s D n ( T , t ) , while the Hankel determinant is related to h n ( T , t ) by the relation D n ( T , t ) = (cid:213) n − j = h j ( T , t ) .We compute the partial derivatives of the Hankel determinant by taking a telescopic sum from j = j = n − h n , i.e. equations (A36) and (A37). Using (A3),60e then obtain ¶ T ( log D n ) = − n − (cid:229) j = R j , (A55) ¶ t ( log D n ) = n − (cid:229) j = R ∗ j . (A56)Hence we now have expressions of the partial derivatives of D n ( T , t ) in terms of the auxiliaryvariables R n and R ∗ n .At this point, recalling equation (44) that H n ( T , t ) = ( T ¶ T + t ¶ t ) log D n ( T , t ) . (A57)This quantity related to the logarithmic derivative of the Hankel determinant is of special interestas our goal is to find the PDE that H n ( T , t ) satisfies. From the definition of H n ( T , t ) , along with(A55) and (A56), we find that H n ( T , t ) = − T n − (cid:229) j = R j + t n − (cid:229) j = R ∗ j , (A58) = p ( n ) + n ( n + a ) , (A59)where the second equality is a result of (A31).At this point, we may express the fundamental quantity H n ( T , t ) in terms of the auxiliary vari-ables and b n . This is given in the following key lemma. Lemma 6. H n ( T , t ) = r n r ∗ n + ( n − N s + a + T ) r n − ( n + N s + a + t ) r ∗ n − ( + R n ) r ∗ n ( r ∗ n − N s ) R ∗ n + ( − R ∗ n ) r n ( r n − N s ) R n + b n R n − b n R ∗ n . (A60) Proof.
Using (A28) and (A29), we obtain the sum t n − (cid:229) j = R ∗ j − T n − (cid:229) j = R j in closed form. All that remains is to eliminate R ∗ n − and R n − from this equation. We do thisusing (A25) and (A26). 61o continue, we apply t ¶ T + t ¶ t to equation (A59), and find that ( T ¶ T + t ¶ t ) H n = p ( n ) + b n , (A61)where we have used (A42) to replace ( T ¶ T + t ¶ t ) p ( n ) by p ( n ) + b n . We see that equations (A59) and (A61) can be regarded as a set of simultaneous equations for p ( n ) and b n . Solving for p ( n ) and b n , we find that p ( n ) = H n − n ( n + a ) , and b n = ( T ¶ T + t ¶ t ) H n − H n + n ( n + a ) . (A62)Before completing the proof of Theorem 1, we show here that our Hankel determinant D n ( T , t ) is related to the t -function of a Toda PDE.Substituting the definition (44) into (A62) and together with (A9), we find ( T ¶ T + t ¶ t ) log D n − ( T ¶ T + t ¶ t ) log D n + n ( n + a ) = D n + D n − D n . Defining ˜ D n ( T , t ) : = ( T t ) − n ( n + a ) D n ( T , t ) , after some computations, ˜ D n ( T , t ) satisfies the following equation: (cid:18) Tt ¶ T T + ¶ Tt + tT ¶ tt (cid:19) log ˜ D n ( T , t ) = ˜ D n + ˜ D n − ˜ D n . (A63)The above equation is a two parameter generalization of the Toda molecule equation, and hencewe identify ˜ D n ( T , t ) as the corresponding t -function of the two-parameter Toda equations.A reduction may be obtained by writing ˜ D n ( T , t ) as˜ D n ( T , t ) = (cid:18) Tt (cid:19) n ( n + a ) F n ( T ) , and equation (A63) is reduced to a 1-parameter Toda equation, ¶ T T log F n ( T ) = F n + ( T ) F n − ( T ) F n ( T ) . artial Differential Equation for H n ( T , t ) We differentiate (A59) with respect to T and t , and make use of the expressions for ¶ T p ( n ) and ¶ t p ( n ) , i.e., (A40) and (A41) respectively, to give ¶ T H n = r n , (A64) ¶ t H n = − r ∗ n . (A65)Thus, we now have expressed r n and r ∗ n in terms of the partial derivatives of H n .Next we derive representations of R n and R ∗ n in terms of H n and its partial derivatives.The idea, following from previous work, is to express R n and 1 / R n in terms H n and itsderivatives w.r.t. t and T . Similarly, for R ∗ n and 1 / R ∗ n . We start by re-writing (A26) as b n R n − = r n ( r n − N s ) R n , and substitute into (A38) to arrive at ¶ T b n = r n ( r n − N s ) R n − b n R n , which is a linear equation in R n and 1 / R n .Substituting r n given by (A64) and b n given by (A62) into the above equation produces ( T ¶ T T + t ¶ Tt ) H n = ( ¶ T H n ) (cid:0) ¶ T H n − N s (cid:1) R n − (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17) R n . (A66)Going through a similar process we find, using (A25), (A62) and (A65), ( T ¶ Tt + t ¶ tt ) H n = − ( ¶ t H n ) (cid:0) ¶ t H n + N s (cid:1) R ∗ n + (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17) R ∗ n . (A67)The above two equations are quadratic equations in R n and R ∗ n . Solving for them leads to2 (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17) R n = − ( T ¶ T T + t ¶ Tt ) H n ± A ( H n ) , (A68)2 (cid:16) T ( ¶ T H n ) + t ( ¶ t H n ) − H n + n ( n + a ) (cid:17) R ∗ n = ( T ¶ Tt + t ¶ tt ) H n ± A ( H n ) , (A69)where A ( H n ) and A ( H n ) are defined by (47) and (48) respectively, where we have left out thenotation that indicates that A and A also depends upon the partial derivatives of H n .In the last step we substitute (A64), (A65), (A62), R n given by (A68) and R ∗ n given by (A69)into (A60). After some simplification, we obtain the PDE (46), completing the proof of Theorem1. 63 ppendix B: Some Relevant Integral Identities The following integral identities, taken from Refs. 13 and 49, are referenced for use in theCoulomb fluid derivations.Let W ( t ) = p ( t + a )( t + b ) : b Z a dx p ( b − x )( x − a ) = p , (B1) b Z a dx ( x + t ) p ( b − x )( x − a ) = p W ( t ) , (B2)12 p b Z a xdx p ( b − x )( x − a ) = ( a + b ) , (B3) P b Z a p ( b − y )( y − a )( y − x )( y + t ) dy = p (cid:18) W ( t ) x + t − (cid:19) , (B4)12 p b Z a log ( x + t ) p ( b − x )( x − a ) dx = log (cid:18) √ t + a + √ t + b (cid:19) , (B5)12 p b Z a log ( x + t ) x p ( b − x )( x − a ) dx = √ ab log (cid:16) √ ab + W ( t ) (cid:17) − t (cid:16) √ a + √ b (cid:17) , (B6)12 p b Z a x log ( x + t ) p ( b − x )( x − a ) dx = (cid:16) √ t + a − √ t + b (cid:17) , + ( a + b ) (cid:16) W ( t ) + t (cid:17) − ab t , (B7)12 p b Z a log ( x + t )( x + t ) p ( b − x )( x − a ) dx = − W ( t ) log (cid:18) √ t + a + √ t + b (cid:19) , (B8) Z dxx √ a ′ x + b ′ x + c ′ = − √ c ′ log c ′ + b ′ x + √ c ′ √ a ′ x + b ′ x + c ′ x ! , (B9)where ( c ′ > ) , = √ c ′ log (cid:18) x c ′ + b ′ x (cid:19) , where ( c ′ > , b ′ = a ′ c ′ ) . (B10) ( A + B ) = log A + Z Bd h A + h B . (B11) Lemma 7.
From the above integrals, we have p b Z a log ( x + t )( x + t ) p ( b − x )( x − a ) dx = W ( t ) log (cid:16) W ( t ) + W ( t ) (cid:17) − ( t − t ) (cid:16) √ t + a + √ t + b (cid:17) . (B12) Proof.
We rewrite log ( x + t ) using (B11) and then use (B2) to obtain LHS ( B ) = log t p ( t + a )( t + b ) + p Z b Z a xdxd h ( t + x h )( x + t ) p ( b − x )( x − a ) . Now we use the partial fraction decomposition x ( t + x h )( x + t ) = t ( t h − t )( x + t ) − t ( t h − t )( t + x h ) , and then integrate over the x variable using (B2) to get LHS ( B ) = log t p ( t + a )( t + b )+ Z d h t h − t t p ( t + a )( t + b ) − t p ( t + a h )( t + b h ) ! , = log ( t − t ) p ( t + a )( t + b ) −
12 lim e → ± e Z t d h ( t h − t ) p ( t + a h )( t + b h ) . (B13)In our problem t = t ′ , t = T ′ or t = T ′ , t = t ′ , hence t / t = ( t ′ / T ′ ) ± = ( t / T ) ± = ( + cs ) ± . For the case s =
0, there exists another pole within the integrand, and so we replace the R . . . with ± e R . . . so that we may invoke (B9).To evaluate the remaining integral in (B13), we first make the change of variable y = t h − t ,giving
12 lim e → ± e Z t d h ( t h − t ) p ( t + a h )( t + b h ) =
12 lim e → t − t ± e t Z − t ( t / t ) dyy p ( t ( t + a ) + ay )( t ( t + b ) + by ) . (B14) Within the square root term, we have a quadratic function in y , given by aby + (cid:16) ( a + b ) t t + abt (cid:17) y + t ( t + a )( t + b ) . a and b are given by (83) and b ≥
0, we see that the discriminant of thequadratic form is given by t ( b − a ) > , while the constant term is t (cid:16) t + t ( + b ) + b (cid:17) > . Hence we may invoke (B9) and then take the limit e → RHS ( B ) = − p ( t + a )( t + b ) − log ( t − t )+ log p ( t + a )( t + b ) p ( t + a )( t + b ) + ab + ( t + t )( a + b ) + t t p ( t + a )( t + b ) + t + a + b ! ! . (B15) Substituting this back into (B13) and after some algebra, we get (B12).
Appendix C: Differential Equations For Large Scale Corrections To Cumulants1. k ( t ′ ) The correction terms f i ( t ′ ) for i = , , , (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) d f ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C1) (cid:0) ( t ′ + b ) + t ′ (cid:1) N s t ′ ( + b ) d f ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C2) (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) d f ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C3) (cid:0) ( t ′ + b ) + t ′ (cid:1) N s t ′ ( + b ) d f ( t ′ ) dt ′ = l ( ) ( t ′ ) . (C4) The functions l ( i ) ( t ′ ) , i = , , , l ( ) ( t ′ ) = t ′ − ( b + ) t ′ − (cid:0) b + b + (cid:1) t ′ − ( b + ) b t ′ + b . (C5) l ( ) ( t ′ ) = − t ′ − ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ + ( b + ) (cid:0) b − b − (cid:1) t ′ − b (cid:0) b − b − (cid:1) t ′ − b ( b + ) t ′ + b . (C6) l ( ) ( t ′ ) = t ′ − ( b + ) t ′ − (cid:0) b − b − (cid:1) t ′ − ( b + ) (cid:0) b − b − (cid:1) t ′ + (cid:0) b − b − b + b + (cid:1) t ′ + ( b + ) (cid:0) b − b − (cid:1) t ′ b + b (cid:0) b + b + (cid:1) t ′ ( b + ) b t ′ + b . (C7) l ( ) ( t ′ ) = − t ′ + ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ + ( b + ) (cid:0) b − b − (cid:1) t ′ − (cid:0) b − b − b + b + (cid:1) t ′ − ( b + ) (cid:0) b − b − b + b + (cid:1) t ′ − b (cid:0) b + b + b − b − (cid:1) t ′ + b ( b + ) (cid:0) b − b − (cid:1) t ′ + b (cid:0) b + b + (cid:1) t ′ − b ( b + ) t ′ + b . (C8) k ( t ′ ) The correction terms g i ( t ′ ) for i = , , , (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) dg ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C9) (cid:0) ( t ′ + b ) + t ′ (cid:1) N s t ′ ( + b ) dg ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C10) (cid:0) ( t ′ + b ) + t ′ (cid:1) / t ′ ( + b ) dg ( t ′ ) dt ′ = l ( ) ( t ′ ) , (C11) (cid:0) ( t ′ + b ) + t ′ (cid:1) N s t ′ ( + b ) dg ( t ′ ) dt ′ = l ( ) ( t ′ ) . (C12) The terms l ( i ) ( t ′ ) , i = , , , l ( ) ( t ′ ) = − t ′ + ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ − (cid:0) b + b + (cid:1) ( b + ) t ′ − (cid:0) b + b + (cid:1) t ′ b − ( b + ) t ′ b + b + t ′ N s h − t ′ + ( b + ) t ′ + t ′ b + ( b + ) t ′ b − (cid:0) b − b − (cid:1) t ′ b − ( b + ) b i . (C13) l ( ) ( t ′ ) = t ′ − ( b + ) t ′ − (cid:0) b − b − (cid:1) t ′ − ( b + ) (cid:0) b + b + (cid:1) t ′ + (cid:0) b − b − b − b − (cid:1) t ′ + b ( b + ) (cid:0) b − b − (cid:1) t ′ − b (cid:0) b − b − (cid:1) t ′ − b ( b + ) t ′ + b . (C14) l ( ) ( t ′ ) = − t ′ + ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ − ( b + ) (cid:0) b − b − (cid:1) t ′ − (cid:0) b − b − b − b − (cid:1) t ′ ( b + ) (cid:0) b − b − b − b − (cid:1) t ′ + b (cid:0) b − b − b + b + (cid:1) t ′ + b ( b + ) (cid:0) b − b − (cid:1) t ′ + b (cid:0) b + b + (cid:1) t ′ − b ( b + ) T + b + t ′ N s h − t ′ + ( b + ) t ′ + (cid:0) b − b − (cid:1) t ′ + ( b + ) (cid:0) b − b − (cid:1) t ′ − (cid:0) b − b − b + b + (cid:1) t ′ − ( b + ) (cid:0) b − b − b + b + (cid:1) t ′ − b (cid:0) b + b + b − b − (cid:1) t ′ + b ( b + ) (cid:0) b − b − (cid:1) t ′ + b (cid:0) b + b + (cid:1) t ′ − b ( b + ) i . (C15) l ( ) ( t ′ ) = t ′ − ( b + ) t ′ − (cid:0) b − b − (cid:1) t ′ − ( b + ) (cid:0) b + b + (cid:1) t ′ + (cid:0) b − b − b + b + (cid:1) t ′ + ( b + ) (cid:0) b − b − b + b + (cid:1) t ′ − b (cid:0) b − b − b + b + (cid:1) t ′ − b ( b + ) (cid:0) b − b − b + b + (cid:1) t ′ − b (cid:0) b + b − b − b − (cid:1) t ′ + b ( b + ) (cid:0) b − b − (cid:1) t ′ + b (cid:0) b + b + (cid:1) t ′ − b ( b + ) t ′ + b . (C16) REFERENCES A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity–Part I: System descrip-tion,” IEEE Trans. Commun. , 1927–1938 (2003). A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity–Part II: Implementationaspects and performance analysis,” IEEE Trans. Commun. , 1939–1948 (2003). J. N. Laneman and G. W. Wornell, “Distributed space-time coded protocols for exploiting co-operative diversity in wireless networks,” IEEE Trans. Inform. Theory , 2415–2425 (2003),special issue on space-time transmission, reception, coding and signal processing.68 J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks:efficient protocols and outage behavior,” IEEE Trans. Inform. Theory , 3062–3080 (2004). R. U. Nabar, H. Bolcskei, and F. W. Kneubuhler, “Fading relay channels: Performance limitsand space-time signal design,” IEEE J. Sel. Areas Commun. , 1099–1109 (2004). Y. Fan and J. Thompson, “MIMO configurations for relay channels: Theory and practice,”IEEE Trans. Wireless Commun. , 1774–1786 (2007). P. Dharmawansa, M. R. McKay, and R. K. Mallik, “Analytical performance ofamplify-and-forward MIMO relaying with orthogonal space-time block codes,”IEEE Trans. Commun. , 2147–2158 (2010). Y. Song, H. Shin, and E. Hong, “MIMO cooperative diversity with scalar-gain amplify-and-forward relaying,” IEEE Trans. Commun. , 1932–1938 (2009). G. Szeg¨o,
Orthogonal Polynomials (American Mathematical Society, New York, 1939) pp.ix+401, American Mathematical Society Colloquium Publications, v. 23. C. A. Tracy and H. Widom, “Fredholm determinants, differential equations and matrix models,”Comm. Math. Phys. , 33–72 (1994). A. P. Magnus, “Painlev´e-type differential equations for the recurrence coefficients ofsemi-classical orthogonal polynomials,”
Proceedings of the Fourth International Sym-posium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992) ,J. Comput. Appl. Math. , 215–237 (1995). Y. Chen and G. Pruessner, “Orthogonal polynomials with discontinuous weights,”J. Phys. A , L191–L198 (2005). Y. Chen and M. R. McKay, “Coulumb fluid, Painlev´e transcendents, and the information theoryof MIMO systems,” IEEE Trans. Inform. Theory , 4594–4634 (2012). Y. Chen and A. Its, “Painlev´e III and a singular linear statistics in Hermitian random matrixensembles. I,” J. Approx. Theory , 270–297 (2010). Y. Chen and M. E. H. Ismail, “Ladder operators and differential equations for orthogonal poly-nomials,” J. Phys. A , 7817–7829 (1997). Y. Chen and M. E. H. Ismail, “Jacobi polynomials from compatibility conditions,”Proc. Amer. Math. Soc. , 465–472 (electronic) (2005). Y. Chen and M. V. Feigin, “Painlev´e IV and degenerate Gaussian unitary ensembles,”J. Phys. A , 12381–12393 (2006). 69 E. Basor, Y. Chen, and L. Zhang, “PDEs satisfied by extreme eigenvalues distributions of GUEand LUE,” Random Matrices Theory Appl. , 1150003–1–1150003–21 (2012). E. Basor, Y. Chen, and T. Ehrhardt, “Painlev´e V and time-dependent Jacobi polynomials,”J. Phys. A , 015204, 25 (2010). E. Basor, Y. Chen, and N. Mekareeya, “The Hilbert series of N = SO ( N c ) and Sp ( N c ) SQCD,Painlev´e VI and integrable systems,” Nuclear Phys. B , 421–463 (2012). Y. Chen and N. Lawrence, “On the linear statistics of Hermitian random matrices,”J. Phys. A , 1141–1152 (1998). F. J. Dyson, “Statistical theory of the energy levels of complex systems. I,”J. Mathematical Phys. , 140–156 (1962). F. J. Dyson, “Statistical theory of the energy levels of complex systems. II,”J. Mathematical Phys. , 157–165 (1962). F. J. Dyson, “Statistical theory of the energy levels of complex systems. III,”J. Mathematical Phys. , 166–175 (1962). Y. Chen and S. M. Manning, “Distribution of linear statistics in random matrix models (metallicconductance fluctuations),” J. Phys.: Condens. Matter , 3039–3044 (1994). G. Szeg¨o, “Hankel forms,” Am. Math. Soc. Transl. (2) , 1–36 (1977). P. Kazakopoulos, P. Mertikopoulos, A. L. Moustakas, and G. Caire, “Liv-ing at the edge: a large deviations approach to the outage MIMO capacity,”IEEE Trans. Inform. Theory , 1984–2007 (2011). M. Jimbo, T. Miwa, and K. Ueno, “Monodromy preserving deformation of linear or-dinary differential equations with rational coefficients. I. General theory and t -function,”Phys. D , 306–352 (1981). M. Jimbo and T. Miwa, “Monodromy preserving deformation of linear ordinary differentialequations with rational coefficients. II,” Phys. D , 407–448 (1981). M. Adler and P. van Moerbeke, “Matrix integrals, Toda symmetries, Virasoro constraints, andorthogonal polynomials,” Duke Math. J. , 863–911 (1995). M. Adler and P. van Moerbeke, “Hermitian, symmetric and symplectic random ensembles: PDEsfor the distribution of the spectrum,” Ann. of Math. (2) , 149–189 (2001). M. Adler, T. Shiota, and P. van Moerbeke, “Random matrices, vertex operators and the Virasoroalgebra,” Phys. Lett. A , 67–78 (1995). 70 Y. Chen and M. E. H. Ismail, “Thermodynamic relations of the Hermitian matrix ensembles,”J. Phys. A , 6633–6654 (1997). H. Jafarkhani,
Space-time coding: theory and practice (Cambridge University Press, 2005). M. Simon and M. Alouini,
Digital Communication over Fading Channels , Vol. 86 (John Wiley& Sons, Inc., 2005). M. R. McKay, A. Zanella, I. Collings, and M. Chiani, “Error probability and SINR analysis ofoptimum combining in Rician fading,” IEEE Trans. Commun. , 676–687 (2009). G. Karagiannidis, D. Zogas, and S. Kotsopoulos, “Statistical proper-ties of the EGC output SNR over correlated Nakagami-m fading channels,”IEEE Trans. Wireless Commun. , 1764–1769 (2004). M. L. Mehta,
Random matrices , 3rd ed., Pure and Applied Mathematics (Amsterdam), Vol. 142(Elsevier/Academic Press, Amsterdam, 2004) pp. xviii+688. K. A. Muttalib and Y. Chen, “Distribution function for shot noise in Wigner-Dyson ensembles,”Internat. J. Modern Phys. B , 1999–2006 (1996). E. L. Basor, Y. Chen, and H. Widom, “Determinants of Hankel matrices,”J. Funct. Anal. , 214–234 (2001). E. L. Basor, Y. Chen, and H. Widom, “Hankel determinants as Fredholm determinants,” in
Random matrix models and their applications , Math. Sci. Res. Inst. Publ., Vol. 40 (CambridgeUniv. Press, Cambridge, 2001) pp. 21–29. M. Tsuji,
Potential theory in modern function theory (Maruzen Co. Ltd., Tokyo, 1959) p. 590. F. D. Gakhov,
Boundary value problems (Dover Publications Inc., New York, 1990) pp.xxii+561, translated from the Russian, Reprint of the 1966 translation. Y. Chen and S. M. Manning, “Asymptotic level spacing of the Laguerre ensemble: a Coulombfluid approach,” J. Phys. A , 3615–3620 (1994). Y. Chen and S. M. Manning, “Some eigenvalue distribution functions of the Laguerre ensemble,”J. Phys. A , 7561–7579 (1996). V. A. Marˆcenko and L. A. Pastur, “Distribution of eigenvalues in certain sets of random matri-ces,” Math. USSR Sb. , 457–483 (1967). E. Katzav and I. P´erez Castillo, “Large deviations of the smallest eigenvalue of the Wishart-Laguerre ensemble,” Phys. Rev. E (3) , 040104, 4 (2010). Z. Wang and G. Giannakis, “A simple and general parameterization quantifying performance infading channels,” IEEE Trans. Commun. , 1389–1398 (2003).71 I. S. Gradshteyn and I. M. Ryzhik,
Table of integrals, series, and products , seventh ed. (Else-vier/Academic Press, Amsterdam, 2007) pp. xlviii+1171, translated from the Russian, Trans-lation edited and with a preface by Alan Jeffrey and Daniel Zwillinger, With one CD-ROM(Windows, Macintosh and UNIX). E. Basor and Y. Chen, “Painlev´e V and the distribution function of a discontinuous linear statisticin the Laguerre unitary ensembles,” J. Phys. A , 035203, 18 (2009). Y. Chen and L. Zhang, “Painlev´e VI and the unitary Jacobi ensembles,”Stud. Appl. Math. , 91–112 (2010). P. J. Forrester and C. M. Ormerod, “Differential equations for deformed Laguerre polynomials,”J. Approx. Theory , 653–677 (2010). J. Shohat, “A differential equation for orthogonal polynomials,”Duke Math. J. , 401–417 (1939). W. C. Bauldry, “Estimates of asymmetric Freud polynomials on the real line,”J. Approx. Theory , 225–237 (1990). S. S. Bonan and D. S. Clark, “Estimates of the Hermite and the Freud polynomials,”J. Approx. Theory , 210–224 (1990). H. N. Mhaskar, “Bounds for certain Freud-type orthogonal polynomials,”J. Approx. Theory , 238–254 (1990). K. Sogo, “Time-dependent orthogonal polynomials and theory of soliton. Applications to matrixmodel, vertex model and level statistics,” J. Phys. Soc. Japan , 1887–1894 (1993). Y. Chen and D. Dai, “Painlev´e V and a Pollaczek-Jacobi type orthogonal polynomials,”J. Approx. Theory162