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The Tracy-Widom distribution is not infinitely divisible. J. Armando Dom´ınguez-Molina Facultad de Ciencias F´ısico-Matemáticas 6 1 Universidad Autónoma de Sinaloa, México 0 2 n Abstract a J The classical infinite divisibility of distributions related to eigenvalues of some ran- 2 1 dom matrix ensembles is investigated. It is proved that the β-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a β-Hermite ] R ensemble, is not infinitely divisible. Furthermore, for each fixed N 2 it is proved ≥ P that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible. . h Keywords: beta Hermite ensembles; random matrices; largest eigenvalue, tail probabilities. t a m [ 1 Introduction 1 v Random matrix theory is an important field in probability, statistics and physics. One of 8 9 the aims of random matrix theory is to derive limiting laws for the eigenvalues of ensembles 8 of large random matrices. In this sense this note will focus in the study the behavior of 2 0 eigenvalues of two types of matrix ensembles, the invariant Hermite and the tridiagonal . 1 β-Hermite. 0 The invariant Hermite ensembles consist of the Gaussian orthogonal, unitary, or symplectic 6 1 ensembles, G(O/U/S)E, which are ensembles of N N real symmetric, complex Hermitian × : or Hermitian real quaternion matrices, H, respectively, whose matrix elements are inde- v i pendently distributed random Gaussian variables with probability density function (PDF) X proportional, modulo symmetries, to r a exp βtrH2 , −4 here, β = 1,2 or 4 is used for the G(O/U(cid:0)/S)E ens(cid:1)embles, respectively. The joint PDF of their ordered eigenvalues λ λ is given by 1 N ≤ ··· ≤ N k λ λ βexp β λ2 , (1) N,β | i − j| −4 i ! 1≤i<j≤N i=1 Y X where k is a non negative constant and for β = 1,2 or 4, it can be computed by Selberg’s N,β Integral formula (see [2, Theorem 2.5.8]). The PDF (1) exhibits strong dependence of the 1 eigenvalues of the G(O/U/S)E ensembles. For more details related to these ensembles see [17], [9], [13], [2, sections 2.5 and 4.1 ]. The law (1) has a physical sense since it describes a one-dimensional Coulomb gas at inverse temperature β, [13, Section 1.4]. Each member of the G(O/U/S)E ensembles leads, by the Householder reduction, to a symmetric tridiagonal matrix (Hβ) of the form N N≥1 N (0,2) χ (n−1)β χ N (0,2) χ 1  (n−1)β (n−2)β  Hβ := ... ... ... , (2) N √β    χ N (0,2) χ  2β β    χ N (0,2)  β     where χ is the χ-distribution with t degrees of freedom, whose probability density function t is given by f (x) = 21−t/2xt−1e−x2/2/Γ(t/2). Here, Γ(α) = ∞vα−1e−vdv is Euler’s Gamma t 0 function. The matrix (2) has the important characteristic that all entries in the upper trian- R gular part are independent. Trotter [35] apply the Householder reduction for the symmetric case (β = 1) while for the unitary and symplectic cases (β = 2,4) the former reduction was applied by Dumitriu and Edelman [11]. Furthermore the late considers the ensemble (2) for general β > 0 proving that in this case the PDF of the ordered eigenvalues of Hβ is N still the PDF (1), see [1, Chapter 20] and [2, Section 4.1]. This matrix model it will named β-Hermite ensemble. The classical Tracy-Widom distribution is defined as the limit distribution of the largest eigenvalue of a G(O/U/S)E random matrix ensemble. It is important due to its applica- tionsinprobability, combinatorics, multivariatestatistics, physics, amongotherapplications. Tracy and Widom [33], [34] have written concise reviews for the situations where their distribution appears. The β-Tracy-Widom distribution is defined as the limiting distribution of the largest eigenvalue of a β-Hermite ensemble. In the case β = 1,2, 4 the β-Tracy-Widom distribution coincides with the classical Tracy-Widom, [21]. The main purpose of this paper is to determine the infinite divisibility of the classical Tracy- Widom and β-Tracy-Widom distributions as well as the infinite divisibility of the largest eigenvalue of the finite dimensional random matrix of GOE and GUE ensembles. Recall that a random variable X is said to be infinitely divisible if for each n 1, there ≥ exist independent random variables X ,...,X identically distributed such that X is equal 1 n in distribution to X + + X . This is an important property from the theoretical and 1 n ··· applied point of view, since for any infinitely divisible distribution there is an associated Lévy process; Sato [24], Rocha-Arteaga and Sato [23]. These jump processes have been recently used for modelling purposes in a broad variety of different fields, including finance, insurance, physics among others; see Barndorff-Nielsen et al. [5], Cont and Tankov [8] and 2 Podolskij et al. [20] and for a physicists point of view see Paul and Baschnagel [19]. Other applications concern deconvolution problems in mathematical physics, Carasso [7]. This note is structured as follows: in section 2 it is presented preliminary results on the tail behavior of the classical and generalized β-Tracy-Widom distribution, useful to analyze infinite divisibility. The non infinite divisibility of the classical and generalized β-Tracy- Widom distribution is proved in Section 3. In Section 4 it is shown that for each N 2 the ≥ largest eigenvalue of a G(O/U)E ensemble is not infinitely divisible. Finally in Section 5 it is presented more new results and some open problems. 2 Tracy-Widom distributions 2.1 Classical Tracy-Widom distribution It is well known that the unique possible limit distributions for the maximum of independent random variables are the Gumbel, Fréchet and Weibull distributions. To classify the limit laws for the maximum of a large number of non independent random variables is still open problem. A possible strategy is to deal with particular models of non independent random variables. Theeigenvalues ofrandommatrices provideagoodexample ofsuch nonindependent random variables. For the Gaussian ensembles, i.e. for N N matrices with independent Gaussian × entries, the joint density function of their eigenvalues, λ λ is given by (1), and 1 N ≤ ··· ≤ because the Vandermonde determinant λ λ β they are strongly dependent. 1≤i<j≤N | i − j| Due to the non independence of random variables with density function (1) it follows that Q the limit distribution of λ = λ is not an usual extreme distributions. The distribution max N of λ converges in the limit N to the Tracy–Widom laws. max → ∞ Tracy-Widom, [31], [32] proved that the following limit, which is denoted by F , exists β F (x) := lim P N1/6 λ 2√N x ,β = 1,2,4 β max N→∞ − ≤ h (cid:16) (cid:17) i and in this case ∞ F (x) = exp (s x)[q(s)]2ds , 2 − − (cid:18) Zx (cid:19) where q is given in terms of the solution to a Painlevé type II equation, and 1/2 1/2 F (x) = exp[E(x)]F (x), F (x) = cosh[E(x)]F (x), 1 2 4 2 where E(x) = 1 ∞q(s)ds. −2 x Furthermore it can be deduced from [31], [32], [34] and [4] (see also [2, Exercise 3.9.36]) that R the asymptotics for F (x) as x , for β = 1,2 or 4 is, β → ∞ 1 F ( x) = exp βx3[1+o(1)] , (3) β − −24 (cid:26) (cid:27) 3 2 1 F (x) = exp βx3/2[1+o(1)] , (4) β − −3 (cid:26) (cid:27) where o(1) is the little-o of 1 which means that lim o(1) = 0. x→∞ With the tail probabilities (3) and (4) it is possible to conclude that the Tracy-Widom distribution is not infinitely divisible for β = 1,2,4 using the results of Section 4. Nevertheless it is convenient go to next section from which it will arrive at the non infinite divisibility of β-Tracy-Widom distribution for any β > 0, by considering the limit distribution of the largest eigenvalue of a matrix Hβ defined in (2) for any β > 0. N 2.2 β-Tracy-Widom distribution The tridiagonal β-Hermite ensemble (2) can be considered as a discrete random Schro¨dinger operator. This stochastic operator approach to random matrix theory was conjectured by EdelmanandSutton[12],andwasprovedbyRam´ırez, RiderandVirág[21], whoinparticular established convergence of the largest eigenvalue of a β-Hermite ensemble for any β > 0. Let λ = λ Hβ , with Hβ defined as in (2), in [21] it is shown the existence of a max N N N β-Tracy-Widom(cid:16)rand(cid:17)om variable TWβ such that N1/6 λ 2√N d TW , max β − N−→→∞ (cid:16) (cid:17) where the β-Tracy-Widom random variable is identified through a random variational prin- ciple: ∞ ∞ TW := sup 2 f2(x)db(x) (f′(x))2 +xf2(x) dx , β β − f∈L(cid:26) Z0 Z0 (cid:27) h i in which x b(x) is a standard Brownian Motion and L, is the space of functions f which satisfy f (0)→= 0, ∞f2(x)dx = 1, ∞ (f′(x))2 +xf2(x) dx < . 0 0 ∞ R R (cid:2) (cid:3) Thecasesβ = 1,2,4coincidewiththeclassicalTracy-WidomdistributionF (x) = P (TW x), β β ≤ [21]. Ram´ırez, et al. [21] also proved that the tails of TW are given by (3) and (4) for any β β > 0. 3 Non infinite divisibility of Tracy-Widom distributions First recall a well known result on a characterization of the Gaussian distribution in terms of the tail behavior; see [30, Corollary 4.9.9]: a non-degenerate infinitely divisible random variable X has a normal distribution if, and only if, it satisfies limsup logP ( X > x) − | | = . (5) x xlogx ∞ → ∞ 4 Now, from (3) and (4) we get the following lemma, where, as usual, the expression f (s) ∼ g(s) means that f (s)/g(s) tend to 1 when s . −→ ∞ Lemma 1 (Two sided tails of the β-Tracy-Widom distribution) Let β > 0, let TW be a β random variable β-Tracy-Widom distributed. Then when x , it follows that → ∞ P ( TW > x) P (TW > x) = exp 2βx3/2[1+o(1)] . | β| ∼ β −3 Proof. Using (3) and (4) we get (cid:0) (cid:1) P ( TW > x) = P (TW < x)+1 P (TW < x) β β β | | − − 1 = exp βx3(1+o(1)) +exp 2βx3/2[1+o(1)] , −24 −3 (cid:18) (cid:19) (cid:0) (cid:1) and hence P ( TW > x) 1 lim | β| = 1+ lim exp βx3(1+o(1))+ 2βx3/2[1+o(1)] x→∞ exp 2βx3/2[1+o(1)] x→∞ −24 3 −3 (cid:18) (cid:19) = 1. (cid:0) (cid:1) Finally, using Lemma 1 it is possible to conclude that the β-Tracy-Widom distribution is not infinite divisibility: Theorem 2 For any β > 0 the β-Tracy Widom distribution is not infinitely divisible. Proof. Let assume that X is infinitely divisible and given that neither it is normal nor degenerate we must have that (5) is false, that is logP ( X > x) lim − | | < . x→∞ xlogx ∞ However, logP ( X > x) 1 P(|X|>x)exp −2βx3/2[1+o(1)] lim − | | = lim − log (cid:16) 3 (cid:17) x→∞ xlogx x→∞ xlogx exp −2βx3/2[1+o(1)] (cid:18) (cid:16) 3 (cid:17) (cid:19) 1 = lim log P(|X|>x) + 2βx3/2[1+o(1)] x→∞ xlogx − exp −2βx3/2[1+o(1)] 3 (cid:26) (cid:18) (cid:16) 3 (cid:17)(cid:19) (cid:27) 2β√x = lim 3 [1+o(1)] x→∞ logx = , ∞ the third equality follows from Lemma 1. Remark 3 Taking β = 1,2 or 4 in Theorem 2 the non infinite divisibility of the classical Tracy-Widom distribution is deduced. 5 4 Non infinite divisibility in the finite N case The following Lemma is necessary to determine the non infinite divisibility of the largest eigenvalue of a random matrix of a GOE/GUE ensemble. Lemma 4 If X is a non Gaussian real random variable such that P ( X > x) ae−bxc with a,b > 0, c > 1, | | ≤ then X is not infinitely divisible. Proof. As in the proof of Theorem 2 it is only necessary to prove that X satisfy the limit (5). Indeed, logP ( X > x) log ae−bxc loga+bxc lim − | | lim − = lim − = . x→∞ xlogx ≥ x→∞ xlogx x→∞ xlogx ∞ (cid:0) (cid:1) Consider, λN the largest eigenvalue of a GOE ensemble of dimension N. In [3, Lemma 6.3] max is proved that the two sided tails of the largest eigenvalue of a GOE satisfy the following inequality P λN x e−Nx2/9 max ≥ ≤ andifλN thelargesteigenvalueofa(cid:0)(cid:12)GUE(cid:12)ensem(cid:1)bleofdimensionN.Thefollowinginequality max (cid:12) (cid:12) is deduced in [15] P λN E λN x 2e−2Nx2. max − max ≥ ≤ from which, with help of Lem(cid:0)m(cid:12) a 4, the n(cid:0)ext t(cid:1)h(cid:12)eorem(cid:1) follows: (cid:12) (cid:12) Theorem 5 Let λN be the largest eigenvalue of a random matrix of a GOE/GUE ensemble max of random matrices. For all N 2, λN is not infinitely divisible. ≥ max Remark 6 The case N = 2 follows from Dom´ınguez-Molina and Rocha-Arteaga [10]. 5 Discussion Recallthataninfinitelydivisiblerandomvariable,X,inR mustcomplythat logP (X > x) + − ≤ axlogx, for some a > 0 and x sufficiently large, [29]. With this result it is possible to deduce the non infinite divisibility of the following random variables: I) Wigner surmise: P (s) = πsexp πs2 . 2 −4 II) The absolute value of TW , Y = TW . β β β (cid:0) | (cid:1) | III) The truncation to the left or to the right of TW . β Open problems 6 1. Free infinite divisibility of the classical Tracy-Widom distribution or the general β- Tracy-Widom distribution. 2. For each N 2, the non infinite divisibility of the largest eigenvalue of random matrix ≥ of a GSE ensemble. 3. Determine if the Tracy-Widom distribution is indecomposable. 4. Investigate the infinite divisibility of λ Hβ in the tridiagonal β-Hermite ensemble max N (2). The article [16] it may be useful. (cid:16) (cid:17) 5. Look for an infinitely divisible interpolation between Tracy-Widom distribution and other distribution (except in the Tracy Widom case). Johanson [14] discuss interpolation between the Gumbel distribution, and the Tracy-Widom distribution. Bohigas et al [6] discuss a continuous transition from Tracy-Widom distribution to the Weilbull distribution, and from Tracy-Widom distribution to Gaussian distribution. It is not known if these interpolations are infinitely divisible (except in the Tracy Widom case). References [1] Akemann, G. , Baik, J. and Di Francesco, P. (2011). The Oxford Handbook of Random Matrix Theory. Oxford Handbooks in Mathematics [2] Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge University Press. [3] Arous, B. G., Dembo, A. and Guionnet, A. (2001). Aging of spherical spin glasses. Probab. Theory Relat. Fields 120, 1–67 [4] Baik, J., Buckingham, R. & DiFranco, J. (2008).Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function. Comm. Math. Phys., 280(2):463– 497, 2008. [5] Barndorff-Nielsen, O.E., Mikosh, T. and Resnick, S. I. (2001), Lévy Processes: Theory and Applications (Birkhaüser, Boston). [6] Bohigas, O., de Carvalho, J. X. and Pato, M. P. (2009) Deformations of the Tracy- Widom distribution. Physical review E 79, 031117. [7] Carasso, A. S. (1992). Infinite divisibility and the identification of singular waveform. In Byrnes, J.S., Hargreaves, K. A. and Berry, K. (1992). Probabilistic and Stochastic Methods in Analysis, with Applications. Springer. (2012 Reprinted) [8] Cont, R. and Tankov, P. (2003), Financial Modelling with Jump Processes (Chapman and Hall). 7 [9] Deift, P. and Gioev, D. (2009). Random Matrix Theory: Invariant Ensembles and Uni- versality. Courant lecture notes 18 [10] Dom´ınguez Molina, J.A. y Rocha Arteaga, A. (2007). On the Infinite Divisibility of some Skewed Symmetric Distributions. Statistics and Probability Letters, 77, 644-648. [11] Dumitriu, I. and Edelman, A. (2002). Matrix models for beta ensembles, J. Math. Phys. 43(11): 5830–5847. [12] Edelma, A. and Sutton, B. D. (2007). From Random Matrices to Stochastic Operators. Journal of Statistical Physics. Volume 127, Issue 6, pp 1121-1165 [13] Forrester, P. J. (2010). Log-Gases and Random Matrices. London Mathematical Society Monographs. Princeton University Press [14] Johansson, K. (2007). From Gumbel to Tracy-Widom. Probab. Theory Relat. Fields 138:75–112 [15] Ledoux, M. (2003). A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices. In Séminaire de Probabilités XXXVII, volume 1832 of Lecture Notes in Mathematics. Paris, Springer. [16] Ledoux, M. and Rider, B. (2010). Small deviations for beta ensembles. Electronic Jour- nal of Probability. Vol. 15 (2010), Paper no. 41, pages 1319–1343. [17] Mehta, M. L. (2004). Random Matrices, Volume 142, Third Edition. Pure and Applied Mathematics. Academic Press [18] Narayanana, R. and Wells, M. T. (2013). On the maximal domain of attraction of Tracy–Widom distribution for Gaussian unitary ensembles. Statistics and Probability Letters 83 2364–2371 [19] Paul, W. and Baschnagel, J. (2013). Stochastic Processes: From Physics to Finance 2nd ed. Springer [20] Podolskij, M., Stelzer, R., Thorbjørnsen, S., and Veraart, A. E. D. (2016). The Fasci- nation of Probability, Statistics and their Applications: In Honour of Ole E. Barndorff- Nielsen. Springer [21] Ram´ırez, J. A., Rider, B. and Virág, B. (2011). Beta ensembles, stochastic Airy spectrum, and a diffusion. Journal of the American Mathematical Society. Volume 24, Num- ber 4, Pages 919–944 [22] Rider, B. and Sinclair, C. D. (2014). Extremal laws for the real ginibre ensemble. The Annals of Applied Probability. Vol. 24, No. 4, 1621–1651. 8 [23] Rocha-Arteaga, A. and Sato, K. (2003), Topics in Infinitely Divisible Distributions and LévyProcesses (AportacionesMatemáticas, Investigación, No.17.SociedadMatema´tica Mexicana). [24] Sato, K. (2013), Lévy Processes and Infinitely Divisible Distributions 2ndEdition(Cam- bridge University Press). [25] Soshnikov, A. (1999). Universality at the edge of the spectrum in Wigner random matrices, Comm. Math. Phys. 207, no. 3, 697-733. [26] Soshnikov, A. (2002). A Note on the Universality of the Distribution of the Largest Eigenvalue in Certain Sample Covariance Matrices. Journal of Statistical Physics 108 (516) 1033-1056. [27] Soshnikov, A. (2006). Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles, in: J. Asch, A. Joye (Eds.), Mathematical Physics of Quantum Mechanics, vol. 690 of Lecture Notes in Physics, Springer Berlin Heidelberg, [28] Soshnikov, A., and Fyodorov, Y. (2005). On the largest singular values of random matrices with independent Cauchy entries, J. Math. Phys. 46, no. 3, 033302, 15 pp. [29] Steutel, F. W. (1979), Infinite Divisibility in Theory and Practice (with discussion), Scand. J. Statist. 6, 57-64. [30] Steutel, F.W., Van Harn, K., (2003). Infinite Divisibility of Probability Distributions on the Real Line. Marcel-Dekker, New York [31] Tracy, C. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159, 151–174. [32] Tracy, C. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Comm. Math. Phys. 177, 727–754. [33] Tracy, C. and Widom, H. (2002). Distribution Functions for Largest Eigenvalues and Their Applications. ICM 2002 Vol. I, pp. 587-596 [34] Tracy, C. A. and Widom, H. (2009). The Distributions of Random Matrix Theory and their Applications. pp. 753-765. in New Trends in Mathematical Physics: Selected con- tributions of the XVth International Congress on Mathematical Physics. (2009). Edited by Sidoravicius. [35] Trotter, H. F. (1984). Eigenvalue distributions of large Hermitian matrices; Wigner’s semicircle law and a theorem of Kac, Murdock, and Szegö, Advances in Mathematics. 54, 67–82. 9