## On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review (2010)

Citations: | 6 - 0 self |

### BibTeX

@MISC{Bornemann10onthe,

author = {F. Bornemann},

title = {On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review},

year = {2010}

}

### OpenURL

### Abstract

Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) β-ensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.

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Citation Context ... Tracy and Widom [52, 53] in their numerical evaluation of F2 (personal communication by Craig Tracy).822 F. Bornemann calculations; a systematic approach is based on Riemann – Hilbert problems, see =-=[16]-=- and [26] for worked out examples. • The Painlevé III representation (2.43), for LUE with parameter α at the hard edge, satisfies [54, eq. (3.1)] σ(x;1)= x 4 α √ − x + O(1) (x →∞). (3.20) 2 • The Pain... |

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Citation Context ...ee, e.g., [11, 12, 40]), the determinantal representation F2(s) =det ( ) I−KAi↾L2 (s,∞) , KAi(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) , (1.6) x − y was spelt out by Forrester [27] and by Tracy and Widom =-=[52]-=-. The search for an analogue to Gaudin’s method remained unsuccessful since there is no solution of the corresponding eigenvalue problem known in terms of classic special functions [38, p. 453]. It wa... |

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Citation Context ...ecent work of Bornemann [5], exclusively based on solving this asymptotic initial value problem. see (5.8a) below. However, the extension of Gaudin’s method to E2(0; s) is fairly straightforward, see =-=[22]-=- for a determinantal formula that is equivalent to (1.2), namely (5.7) with k =0, and [37] for subsequent numerical work. As was pointed out by Odlyzko [41, p. 305], who himself had calculated E2(0; s... |

155 | On orthogonal and symplectic matrix ensembles
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Citation Context ...th largest eigenvalue The cumulative distribution function of the kth largest eigenvalue is, in the soft edge scaling limit, k−1 ∑ Fβ(k; s) = j=0 E (soft) β (j;(s, ∞)). (2.9) The famous Tracy – Widom =-=[56]-=- distributions Fβ(s) aregivenby { Fβ(1; s), β =1orβ =2, Fβ(s) = Fβ(1; √ 2 s), β =4. (2.10)On the numerical evaluation of distributions in random matrix theory 809 2.2. Determinantal representations f... |

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Citation Context ...known for quite some time (see, e.g., [11, 12, 40]), the determinantal representation F2(s) =det ( ) I−KAi↾L2 (s,∞) , KAi(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) , (1.6) x − y was spelt out by Forrester =-=[27]-=- and by Tracy and Widom [52]. The search for an analogue to Gaudin’s method remained unsuccessful since there is no solution of the corresponding eigenvalue problem known in terms of classic special f... |

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Citation Context ...t of GUE, scaled for the fluctuations at the soft edge (that is, the maximum eigenvalue), yields the function F2(s) =P(no levels lie in (s, ∞)). (1.5) Implicitly known for quite some time (see, e.g., =-=[11, 12, 40]-=-), the determinantal representation F2(s) =det ( ) I−KAi↾L2 (s,∞) , KAi(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) , (1.6) x − y was spelt out by Forrester [27] and by Tracy and Widom [52]. The search for a... |

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Citation Context ... The typical choices of intervals with a fixed bound are J =(0,s) orJ =(s, ∞), depending on whether one looks at the bulk or the soft edge of the spectrum.812 F. Bornemann 2.3.1. GUE Tracy and Widom =-=[55]-=- calculated, for the determinant (2.11b) with J = (s, ∞), the representation D (n) ( ∫∞ ) 2 (z;(s, ∞)) = exp − σ(x; z) dx in terms of the Jimbo–Miwa–Okamoto σ-form of Painlevé IV,namely σ 2 xx =4(σ−xσ... |

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Citation Context ...numerical evaluation1of many of these functions on the one hand and shows, on the other hand, that such work facilitates numerical explorations that may lead, in the sense of Experimental Mathematics =-=[10]-=-, to new theoretical discoveries, see the results of Section 6. 1.1. The common point of view The closed analytic solutions alluded to above are based (for deeper reasons or because of contingency) on... |

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Citation Context ...e the matrix kernel determinant (2.16) for the numerical checks of the fundamental equation (6.10) in Section 6. 8.1.1. An example: the joint probability distribution of GUE matrix diffusion Prähofer =-=[44]-=- proved that the joint probability of the maximum eigenvalue of GUE matrix diffusion at two different times is given, in the soft edge scaling limit, by the operator determinant P ( ( ( ) ) ) K0 Kt A2... |

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Citation Context ...g function E1(0; s) of GOE that he represented as the Fredholm determinant of the even sine kernel,On the numerical evaluation of distributions in random matrix theory 805 Jimbo, Miwa, Môri and Sato =-=[36]-=- expressed the Fredholm determinant by ( ∫πs σ(x) Es(0; s) =exp − x dx ) (1.3) in terms of the Jimbo–Miwa–Okamoto σ-form of Painlevé V,namely (xσxx) 2 =4(σ − xσx)(xσx − σ − σ 2 x), 1.1.2. The Tracy – ... |

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65 |
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Citation Context ...ctions [38, p. 453]. It was therefore a major breakthrough when Tracy and Widom [52, 53] derived their now famous representation ( F2(s) =exp ∫∞ − (x − s)u(x) 2 ) dx in terms of the Hastings – McLeod =-=[34]-=- solution u(x) ofPainlevéII,namely s (1.7) uxx =2u 3 + xu, u(x) ≃ Ai(x) (x →∞). (1.8) Subsequent numerical evaluations were then, until the recent work of Bornemann [5], exclusively based on solving t... |

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55 | Barycentric Lagrange interpolation
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Citation Context ...ight 1/2. This formula enjoys perfect numerical stability [35]. If F is real analytic, we have exponential convergence once more, that is ‖F − pm‖∞ = O(ρ −m ) (m →∞) (4.14) for some constant ρ>1 (see =-=[4]-=-). Low order derivatives (such as densities) and integrals (such as moments) can easily be calculated from this interpolant. All that is most conveniently implemented in [3] chebfun package for Matlab... |

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Citation Context ...for some constant ρ>1 (see [4]). Low order derivatives (such as densities) and integrals (such as moments) can easily be calculated from this interpolant. All that is most conveniently implemented in =-=[3]-=- chebfun package for Matlab (see also [21]). 4.6. Examples We illustrate the method with three examples. More about the software that we have written can be found in Section 9. 4.6.1. Distribution of ... |

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Random matrix theory
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Citation Context ...e solved the initial value problem with b− = 12, using a Runge – Kutta method with automatic error and step size control as coded in Matlab’s ode45, whichis essentially the code published in [23] and =-=[24]-=-. The red lines in Figure 1 show the absolute error |v(x) − u(x)| and the corresponding error in the calculation of F2. We observe that the error of v(x) grows exponentially to the left of b− and the ... |

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Citation Context ...s − sk) ∑ ′′m k=0 (−1)k , (4.13) /(s − sk) where the double primes denote trapezoidal sums, i.e., the first and last term of the sums get a weight 1/2. This formula enjoys perfect numerical stability =-=[35]-=-. If F is real analytic, we have exponential convergence once more, that is ‖F − pm‖∞ = O(ρ −m ) (m →∞) (4.14) for some constant ρ>1 (see [4]). Low order derivatives (such as densities) and integrals ... |

34 |
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Citation Context ...t of GUE, scaled for the fluctuations at the soft edge (that is, the maximum eigenvalue), yields the function F2(s) =P(no levels lie in (s, ∞)). (1.5) Implicitly known for quite some time (see, e.g., =-=[11, 12, 40]-=-), the determinantal representation F2(s) =det ( ) I−KAi↾L2 (s,∞) , KAi(x, y) = Ai(x)Ai′ (y) − Ai ′ (x)Ai(y) , (1.6) x − y was spelt out by Forrester [27] and by Tracy and Widom [52]. The search for a... |

30 |
Level spacing distributions and the Bessel kernel
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Citation Context ...rm of Painlevé V,namely (xσxx) 2 = ( σ − xσx − 2σ 2 ) 2 2 x +(2n+ α)σx − 4σx (σx − n)(σx − n − α), (2.41a) Γ(n + α +1) σ(x; z) ≃ z Γ(n)Γ(α +1)Γ(α+2) xα+1 (x → 0). (2.41b) Accordingly, Tracy and Widom =-=[54]-=- obtained, for the determinant (2.39b) of the scaling limit at the hard edge, the representation D (hard) ( 2 (z;(0,s),α)=exp ∫ − 0 s σ(x; z) x ) dx (2.42) in terms of the Jimbo–Miwa–Okamoto σ-form of... |

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Citation Context ...recision, which renders the straightforward approach numerically instable (that is, unreliable in fixed precision hardware arithmetic). To nevertheless obtain a solution that is accurate to 16 digits =-=[45]-=- turned, instead of changing the method, to variable precision software arithmetic (using up to 1500 significant digits in Mathematica) and solved the initial value problem with b− = 200 and appropria... |

26 | Universality of the distribution functions of random matrix theory. In Integrable systems: from classical to quantum
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Citation Context ...in Figure 4.b. Remark To our knowledge, prior to this work, only calculations of the particular cases k = 1 (the largest level) and k = 2 (the next-largest level) have been reported in the literature =-=[20,58]-=-. These calculations were based on the representation (2.27) of the determinant in terms of the Painlevé II equation (2.30). The evaluation of F2(s) =F2(1; s) was obtained from the Hastings – McLeod s... |

25 | Introduction to random matrices - Tracy, Widom - 1993 |

24 |
Interrelationships between orthogonal, unitary and symplectic matrix ensembles
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(Show Context)
Citation Context ...s given by [29, p. 194] E (soft) ⎧ ⎨ lim β (k; J) = n→∞ ⎩ E(n) β lim ( k; √ 2n +2 −1/2 n −1/6 J ) , β =1orβ =2, n→∞ E(n/2) ( √ ) −1/2 −1/6 β k; 2n +2 n J , β =4. (2.4) 4 We follow Forrester and Rains =-=[29]-=- in the choice of the variances of the Gaussian weights. Note that Mehta [38, Chap. 3], such as Tracy and Widom in most of their work, uses w(x) = exp{−βx 2 /2}. However, one has to be alert: from p. ... |

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22 |
Lagrangian interpolation at the Chebyshev points xn,ν = cos(νπ/n), ν = 0(1)n; some unnoted advantages
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Citation Context ...= 64 approximation are correct. F (s) is given on the finite interval [a, b] ands0,...,sm denote the Chebyshev points of that interval, the polynomial interpolant pm(s) ofdegreem is given by Salzer’s =-=[46]-=- barycentric formula ∑ ′′m pm(s) = k=0 (−1)kF (sk)/(s − sk) ∑ ′′m k=0 (−1)k , (4.13) /(s − sk) where the double primes denote trapezoidal sums, i.e., the first and last term of the sums get a weight 1... |

20 | Universality for mathematical and physical systems
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Citation Context ...hout additional structure there would be, in general,804 F. Bornemann only one method: Monte Carlo simulation. However, because of the universality of certain scaling limits (for a review see, e.g., =-=[17]-=-), a family of distinguished distribution functions enters which is derived from highly structured matrix models enjoying closed analytic solutions. These functions constitute a new class of special f... |

20 |
Sur la loi Limite de l’éspacement des valeurs propres d’une matrice aléatoire
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(Show Context)
Citation Context ...itary Ensemble (GUE). 1.1.1. Level spacing function of GUE The large matrix limit of GUE, scaled for level spacing 1 in the bulk, yields the function E2(0; s) =P(no levels lie in (0,s)). (1.1) Gaudin =-=[31]-=- showed that this function can be represented as a Fredholm determinant, namely, 2 E2(0; s) =det ( ) I − Ksin↾L2 (0,s) , Ksin(x, y) =sinc(π(x−y)). (1.2) He proceeded by showing that the eigenfunctions... |

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15 | A determinantal formula for the GOE Tracy-Widom distribution - Ferrari, Spohn - 2005 |

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14 | Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations
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- 2005
(Show Context)
Citation Context ...e Hastings – McLeod solution of (2.30), the representation ( F1(1; s) =exp − 1 2 ( 1 F4(1; s) =cosh 2 ∫ s ∞ ∫ s ∞ ) u(x) dx F2(1; s) 1/2 , ) u(x) dx F2(1; s) 1/2 . (2.32a) (2.32b) More general, Dieng =-=[20]-=- found Painlevé representations of F1(k; s) andF4(k; s). 2.4. Laguerre ensembles Here, we take, on x ∈ (0, ∞) with parameter α>−1, the weight functions 6 • wα(x) =x α exp{−x/2} for β = 1, the Laguerre... |

14 | Matrix kernels for the Gaussian Orthogonal and Symplectic ensembles’, Ann
- Tracy, Widom
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(Show Context)
Citation Context ...es a finite rank matrix kernel (see [38, Chap. 8]) — a formula that is amenable to the numerical methods of Section 4. We confine ourselves to the soft edge scaling limit of this formula which yields =-=[59]-=- E (soft) 4 (k; J) = (−1)k d k! k dzk √ D4(z; J) ∣ , z=1 (2.16a)810 F. Bornemann where the entries of the matrix kernel determinant ( D4(z; J) =det I − z ( S SD 2 IS S∗ ) ↾ L 2 (J)⊕L 2 (J) ) (2.16b) ... |

14 | On asymptotics for the Airy process
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(Show Context)
Citation Context ...v(A2(t), A2(0)) = var(F2) − t + O(t 2 ) (t → 0), (8.6) where F2 denotes the Tracy – Widom distribution for GUE (the numerical value of var(F2) can be found in Table 4); for large t with the expansion =-=[1, 62]-=- cov(A2(t), A2(0)) = t −2 + ct −4 + O(t −6 ) (t →∞), (8.7) where the constant c = −3.542 ··· can explicitly be expressed in terms of the Hastings – McLeod solution (1.8) of Painlevé II. cov(A 2 (t),A ... |

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11 | On the numerical evaluation of Fredholm determinants
- Bornemann
(Show Context)
Citation Context ...terms of the Hastings – McLeod [34] solution u(x) ofPainlevéII,namely s (1.7) uxx =2u 3 + xu, u(x) ≃ Ai(x) (x →∞). (1.8) Subsequent numerical evaluations were then, until the recent work of Bornemann =-=[5]-=-, exclusively based on solving this asymptotic initial value problem. see (5.8a) below. However, the extension of Gaudin’s method to E2(0; s) is fairly straightforward, see [22] for a determinantal fo... |

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10 |
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(Show Context)
Citation Context ... (x) 2 − xAi(x) 2 ) (x→∞). (2.29)On the numerical evaluation of distributions in random matrix theory 813 2.3.2. GOE and GSE in the bulk With σ(x) =σ(x; 1) from (2.26) there holds the representation =-=[2]-=- ( E1(0; s) =exp − 1 2 ( 1 E4(0; s/2) = cosh 2 ∫ 0 πs ∫ 0 πs √ d dx √ d dx σ(x) x σ(x) x dx ) E2(0; s) 1/2 , dx ) E2(0; s) 1/2 . (2.31a) (2.31b) Painlevé representationsforE1(k; s) andE4(k; s) can be ... |

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7 | Asymptotic independence of the extreme eigenvalues of Gaussian unitary ensemble
- Bornemann
(Show Context)
Citation Context ...dependence of the largest two level. Yet another calculation of joint probabilities in the spectrum of GUE (namely, related to the statistical independence of the extreme eigenvalues) can be found in =-=[6]-=-. 17So, this example stretches our numerical methods pretty much to the edge of what is possible. Note, however, that these numerical results are completely out of the reach of a representation by par... |