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1 Acta Polytechnica 57(6): , 217 Czech Technical Univerity in Prague, 217 doi: /ap available online at NEW SPECTRAL STATISTICS FOR ENSEMBLES OF 2 2 REAL SYMMETRIC RANDOM MATRICES Sachin Kumar a,, Zafar Ahmed b a Theoretical Phyic Section, Bhabha Atomic Reearch Centre, Mumbai 4 85, India b Nuclear Phyic Diviion, Bhabha Atomic Reearch Centre, Mumbai 4 85, India correponding author: achinv@barc.gov.in Abtract. We invetigate pacing tatitic for enemble of variou real random matrice where the matrix-element have variou Probability Ditribution Function (PDF: f(x)) including Gauian. For two modification of 2 2 matrice with variou PDF, we derive the pacing ditribution p() of adjacent energy eigenvalue. Neverthele, they how the linear level repulion near = a α where α depend on the choice of the PDF. More interetingly when f(x) = xe x2 (f() = ), we get cubic level repulion near = : p() 3 e 2. We alo derive the ditribution of eigenvalue D(ɛ) for thee matrice. Keyword: real ymmetric matrice;. 1. Introduction Due to matrix mechanic of Heienberg and Method of Linear Combination of Atomic Orbital (LCAO) one can eaily viualize the eigenpectrum of variou ytem with time-reveral ymmetry a reult of diagonalization of a real ymmetric matrix where the matrix element are calculated uing the inter-particle interaction. Mot of the time thi interaction i not known. For example, energy level of variou nuclei are known experimentally but the nuclear interaction i not really known. Random Matrix Theory [1 5] originated by conidering the level pacing tatitic P (S) between eigenvalue of real ymmetric matrice ( ) ( ) a b a + b c R 1 =, R b c 2 = (1) c a b when matrix element a, b, c are random number with the Probability Ditribution Function (PDF) a Gauian: f(x) = 1 2π e x2. The pacing of eigenvalue are given a S 1 4b 2 + (a c) 2 and S 2 b 2 + c 2, repectively. Notice that the S 1, i function of three parameter (a, b, c), wherea S 2 function of jut two (b, c). Notwithtanding thi diparity and the complexity of the multiple integral f(a, b, c)δ[s S(a, b, c)] da db dc (2) the pacing ditribution in the two cae (R 1, R 2 ) turned out to be the ame: P (S) = Se S2. When arranged to yield the average pacing a 1, the normalized pacing ditribution i written a p W () = π 2 2 e π /4, (3) Thi i called the pacing ditribution of Gauian Orthogonal Enemble (GOE) due to the orthogonal ymmetry of real ymmetric matrice. Moreover p W i well known a Wigner ditribution function. d [1 5] that the pacing ditribution of adjacent eigenvalue of N number of n n Gauian random real ymmetric matrice will again be given by (3). Next, Wigner predicted that (3) would eventually repreent the pacing tatitic of neutron-nucleu cattering reonance and nuclear level. Notice that near zero p W () i linear a π/4, thi i called the linear level repulion of adjacent eigenvalue. The rotational invariance and invariance under time-reveral of a ymmetric matrix lie behind the linear level repulion. Conequently, the pacing of nuclear level of ame angular momentum J and parity π indeed diplay [1 5] p() in (3). Wigner urmie i trange but true, each nucleu behave like a matrix of large order. Even much later, in the recent year invetigation on pacing ditribution of enemble random matrice uing 2 2 continue to be an attractive propoition for both ymmetric/hermitian [7, 8] and non-hermitian matrice [9 14]. In thee work [7, 8] one ha taken Gauian ditribution with zero mean and different variance for variou entrie of the matrice and derived a variety of pacing ditribution. Similarly, under Gauian 418

2 vol. 57 no. 6/217 New Spectral Statitic HaL HbL N= N= Figure 1. p() for (a): R 1 and (b): R 2 matrice due to the uniform ditribution of element. The olid line are due to (7) and (1), dahed line repreent the Wigner ditribution (3). Thee two p() are ditinctly different but near = they are linear (α), with α a 1.23 and 1.1, repectively PDF, for enemble of everal 2 2 peudo-ymmetric and peudo-hermitian repreenting Parity-Time-reveral ymmetric ytem the novel expreion of p() have been derived [9 14]. Here, we how that even two modification of real ymmetric 2 2 random matrice yield different p() under the ame probability ditribution function (PDF) f(x). Similarly, one type of matrix under everal non-gauian PDF yield ditinct expreion for p(). However, all the pacing ditribution diplay the linear (α) level repulion near = wherein α depend on the type of PDF and the type of matrix (1) being ued. The quetion of uing a non-gauian probability ditribution to tet Wigner econd conjecture doe not appear to have attracted much attention after [6]. However, it i generally believed that pectral ditribution are inenitive to PDF. Ue of many non-gauian ditribution for finding the probability [15] of occurrence of real eigenvalue for the product of n number of 2 2 real random matrice i worth mentioning. The other intereting pectral tatitic denoted a D(ɛ) i called ditribution of eigenvalue of n n real ymmetric matrice when n i large. Wigner propoed it to be [1 5] D(ɛ) = 2 π 1 ɛ2, ɛ = E/E (4) which i well known a Wigner emicircle law. In cae of large value of n, E i the maximum eigenvalue of the matrix. Due to the hitoric connection of 2 2 matrice in RMT and pecially due to the atonihing amene of p() in cae of GOE for n = 2 and n >> 2, the quetion ariing here i a to what i the analytic form of D(ɛ) in cae of n = 2. Due to the numerical calculation of Porter we know that qualitatively D(ɛ) make an intereting tranition from a bell hape to the emi-circle a the n increae. In cae of n = 2, we collect a large number (N) of matrice and find the mean of poitive eigenvalue to fix E = Ē to obtain D(ɛ) both analytically and by finding their hitogram numerically. The obtained D(ɛ) once again are unlike the Wigner emi-circle law (4) (for n = 2) and our analytic/emi-analytic reult agree excellently with the numerically computed hitogram. In 2, we wih to preent analytic or emi-analytic p() for four non-gauian PDF of element of matrice for two type of 2 2. Thee non-gauian PDF are : Uniform (U), Exponential (E: f(x) = e x, Super-Gauian (SG: f(x) = e x4 ) and Maxwellian (M: f(x) = xe x2 ). In 3, we derive D(ɛ) for R 1 and R 2 and plot them in Figure 4 with their numerically computed hitogram. 2. Enemble of 2 2 real-ymmetric random matrice and their pacing ditribution In thi ection we find P (S) (2) for two real ymmetric matrice R 1 and R 2 for four PDF (U, E, SG, M). By finding the average pacing a S = SP (S) ds/ P (S) S, we then find the normalized pacing ditribution p( = S/ S). We thu conform to the invariance of ditribution in two S and a p()d = P (S)dS Uniform Ditribution: f(x) = 1, x λ, f(x) =, x > λ For the matrix R 1, we have to evaluate λ λ λ δ [ S 4b 2 + (a c) 2] da db dc, (5) λ λ λ 419

3 Sachin Kumar, Zafar Ahmed Acta Polytechnica HaL HbL N=1 5.4 N= Figure 2. The ame a in Figure 1, for Exponential PDF (E: e x ) ariing from emi-analytic expreion (13) and emi-analytic form (15). Here α value are 4.5 and 2.91, repectively. Without a lo of generality we may chooe λ = 1. Let u introduce the tranformation from (a, b, c) to (u, v, w) a u = a c, v = a + c, w = 2b, (5) become A S du 2 dw 2 u u 2 δ(s w 2 + u 2 ) dv, S 2 S du 2 dw 2 u u 2 δ(s w 2 + u 2 ) dv, 2 < S < 2 2, S 2 (6) 2. We find that integral in (6) can be done and P (S) turn out to be a piecewie continuou function given a A S(π S)/4, S 2, S 2 [in 1 (2/S) in 1 ( S 2 4/S)] + S 4 [ S 2 4 2], 2 < S < 2 2,, S 2 (7) 2. For the matrix R 2 (1), λ λ λ λ λ λ δ[s b 2 + c 2 ] da db dc. (8) The a-integral i eparable and it will yield a multiplicative contant. we convert the double integral in b and c in to polar form a b = r co θ, c = r in θ A 1 π/4 r drδ(s r) dθ, S 1 2 π/4 1 co 1 (1/r) r drδ(s r) dθ, 1 < S < 2, (9), S 2. Finally for real ymmetric matrix R 2 (1) when the element are ditributed uniformly over [-1,1], from (9) we get the continuou three piece pacing ditribution function for (, ) (λ = 1) a A πs/2, S 1, P (S) = 2A S[π/4 co 1 (1/S)], 1 < S < 2,, S (1) 2. In Figure 1, we plot p() ariing from analytic reult (7,1) along with the hitogram generated from 1 (= N), 2 2 real ymmetric matrice of the type (a): R 1 and (b): R 2. Near =, they how linear repulion, where α (the coefficient of linearity) i 1.23 and 1.9, repectively. Notice the excellent agreement of olid line with hitogram, the dahed line repreent Wigner ditribution (3) Exponential Ditribution: f(x) = e x Multiple integral in P (S) for R 1 under exponential PDF can be written a 42 e ( a + b + c ) δ [ S 4b 2 + (a c) 2] da db dc. (11)

4 vol. 57 no. 6/217 New Spectral Statitic HaL HbL.8.9 Analytic.6 N=1 5.6 N= Figure 3. The ame a in Figure 1 and 2, for uper-gauian PDF (SG: e x4 ) ariing from the emi-analytic expreion (18) and the analytic one (19). Here α value are 1.3 and.91, repectively. We tranform P (S) into the three dimenional pherical polar co-ordinate uing 2b = r co θ, a = r in θ co φ, c = r in θ in φ a π 2π Crahing the delta function in above, we get a θ, φ integral π/2 π e r( co θ /2+in θ( co φ + in φ )) δ[s rg(θ, φ)]r 2 dr in θ dθ dφ. (12) e S( co θ /2+in θ( co φ + in φ ))/g(θ,φ) S 2 in θ dθ dφ, g[θ, φ) 3 g(θ, φ) = 1 in 2 θ in 2φ, (13) Due to the ymmetry of integrand the domain of integration in (13) have been reduced. The P (S) of R 2 for exponential ditribution can be written a e ( a + b + c ) δ[s b 2 + c 2 ] da db dc. (14) Here the a-integral i eparable and give 1. The remaining double integral can be converted to polar form a S π/2 e S(in θ+co θ) dθ. (15) The integral (13), (15) are further inexpreible in term of known function. p() for thee two cae are plotted in Figure 2, they look imilar though ditinct, notice their linear behaviour near = like Wigner ditribution (dahed line) Super-Gauian ditribution: f(x) = e x4 For R 1 (1), the P (S) integral (2) become e (a4 +b 4 +c 4) δ[s 4b 2 + (a c) 2 ] da db dc. (16) We tranform P (S) into the three dimenional pherical polar co-ordinate uing 2b = r co θ, a = r in θ co φ, c = r in θ in φ a π 2π Crahing the delta function in above, we get a θ, φ integral π/2 π e r4 (co 4 θ/16+in 4 θ(co 4 φ+in 4 φ)) δ[s rg(θ, φ)]r 2 dr in θ dθ dφ. (17) e S4 (co 4 θ/16+in 4 θ(co 4 φ+in 4 φ))/g 4 (θ,φ) S 2 in θ dθ dφ, g(θ, φ) 3 g(θ, φ) = 1 in 2 θ in 2φ. (18) 421

5 Sachin Kumar, Zafar Ahmed Acta Polytechnica For the matrix R 2, the a-integral in P (S) i eparable give a multiplying contant. Then the double integral in b, c i changed to polar form where by crahing the delta function we get an integral S π/2 e S(co4 θ+in 4 θ) dθ = A Se 3S4 /4 2π e (S4 co t)/4 dt = A πs e 3S4 /4 I (S 4 /4), (19) 2 p() correponding to (18) and (19) are plotted in Figure 3 howing linear level repulion near = Maxwellian Ditribution: f(x) = xe x2, x > For R 1 type of real ymmetric matrix P (S) i not imple, however here we would like to how that when the PDF doe not peak at x =, we get highly non-linear behaviour of P (S) near S = For R 2, to thi end we convert the integral (2) to polar form and get S 3 e S2, (2) diplaying nonlinear cubic behaviour S 3 behaviour near S =. In RMT, the PDF of matrix element i uually taken a ymmetric and peaking at x = and one get linear level repulion near =. But when we take the non-ymmetric Maxwellian ditribution (f() = ), we get cubic level repulion near =. Therefore, it would be intereting to ee whether for n n (n large) the cubic level repulion perit. 3. Ditribution of eigenvalue D(ɛ) of 2 2 Gauian random matrice We collect 2N eigenvalue of N 2 2 matrice to find the mean of poitive eigenvalue (Ē) and divide all eigenvalue by Ē and find hitogram D(ɛ). For a large real ymmetric matrix thi ditribution i well known a emi-circle law (4). The ditribution of eigenvalue E 1 (a, b, c) and E 2 (a, b, c) can be obtained analytically a g(e) = A Ē = g(e) de g(e) de, ɛ = Ē E f(a, b, c)[δ(e E 1 ] + δ(e E 2 )] da db dc, g(ɛē), D(ɛ) = (21) g(ɛē)dɛ. Once again f(x) i the PDF of matrix element. Here, E 1,2 = 1 2 (a + c ± (a c) 2 + 4b 2 ) for R 1 and E 1,2 = a ± b 2 + c 2 for R 2. For R 1, g(e) can be obtained from (21), by uing Gauian PDF, defining a + c = u, a c = v and crahing the delta function w.r.t. u. Next, we ue polar co-ordinate v = r co θ and b = r 2 in θ to get 2π g(e) = A e 2E2 coh(2er)e r2 ( 7 co 2θ ) r dr dθ (22) which reduce to a one-dimenional integral g(e) = A e 2E2 r coh(2er)e 7r2 /8 I (r 2 /8) dr. (23) For R 2 with Gauian PDF, we crah the delta function w.r.t. the variable a and ue polar co-ordinate b = r co θ, c = r in θ and we get a imple form g(e) = e E2[ 2 + 2πE erf(e/ 2)e E2 /2 ] /(4 π). (24) Thi function i normalized to 1 in for E (, ), Ē calculated in E (, ) i 4+π 4 = 1.73, conequently, π D(ɛ) = g(ɛ). See the D(ɛ) hitogram in Figure 4 for eigenvalue of N = matrice:(a) R 1 and (b) R 2 where the matrix element are Gauian random number with mean and variance 1. In Figure 4, D(ɛ) ((23) and (24)) matche well with the hitogram. Uually, D(ɛ) i plotted by taking ɛ = E/E m, where E m i the maximum of the eigenvalue and D(ɛ) i tudied for 1 ɛ 1. With regard to thi the x-axi could be caled down to the domain [ 1, 1] to ee that the enemble of 2 2 real matrice defy the emi-circle law which i oberved for real ymmetric matrice of large order. We alo find that D(ɛ) for both R 1 and R 2 are enitive to the PDF of matrix element. 422

6 vol. 57 no. 6/217 New Spectral Statitic D Ε.3 a D Ε.3 b Ε 3 3 Ε Figure 4. Ditribution of eigenvalue D(ɛ) for R 1 (a) and R 2 (b) in (1) under Gauian PDF of matrix element. The olid line (blue) i due to (23) and (24). The hitogram are due to an enemble of N = matrice. 4. Concluion Our analytic and emi-analytic reult on P (S) for two modification of 2 2 real ymmetric matrice in (7), (1), (13), (15), (18) and (19) under variou probability ditribution function and the plotted p() in Figure 1 3 are new and intructive. They all give the linear level repulion a α near = but notably α i not fixed. The Maxwellian PDF (f() = ) of matrix element preent a triking reult wherein the level repulion near = i cubic. It will be further intereting to invetigate pectral ditribution for n n matrice with PDF which are non-ymmetric and vanih at x =. The ditribution of eigenvalue for two real matrice under Gauian PDF obtained in (23) and (24) are alo new and intructive. Acknowledgement S. K. wihe to thank Dr. Shahi C. L. Srivatava, VECC, Kolkata, for ome clarification on RMT. Reference [1] C. E. Porter, Statitical Theorie of Spectra: Fluctuation (Academic, New York, 1965). [2] M. L. Mehta, Random Matrice 3rd Ed. (Ameterdam, Elevier, 24). [3] A. Bohr and Mottelon, Nuclear Structure vol. I (Benjamin, Reading, MA, 1975). [4] F. Hake, Quantum Signature of Chao (New York Springer, 1992). [5] E.P. Wigner, Ann. Math [6] N. Roenzweig, Phy. Rev. Lett (1958). [7] P.C. Huu-Tai, N. A. Smirnova, P. Van Iacker, J. Phy. A: Math. Gen. 35 L199 (22). [8] M.V. Berry and P. Shukla, J. Phy. A: Theor (29). [9] S. Groman and M. Robnik, J. Phy. A: Math. Theor (27). [1] Z. Ahmed, Phy. Lett. A (23). [11] Z. Ahmed and S.R jain, J. Phy. A: Math. Gen (23). [12] Z. Ahmed and S.R. Jain, Phy. Rev. E (R) (23). [13] E.M. Graefe, S. Mudute-Ndumbe and M. Taylor, J. Phy. Theor FT2 (215). [14] J. Gong and Q. Wang, J. Phy , (212). [15] S. Hameed. K. Jain and A. Laxminaraynan, J. Phy. A: Math. Theor (215). 423

ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION. Xiaoqun Wang

Proceeding of the 2008 Winter Simulation Conference S. J. Maon, R. R. Hill, L. Mönch, O. Roe, T. Jefferon, J. W. Fowler ed. ON THE APPROXIMATION ERROR IN HIGH DIMENSIONAL MODEL REPRESENTATION Xiaoqun Wang