## 2008, The Airy 1 process is not the limit of the largest eigenvalue in GOE matrix diffusion

Citations: | 5 - 3 self |

### BibTeX

@MISC{Bornemann_2008,the,

author = {Folkmar Bornemann and Patrik L. Ferrari and Michael Prähofer},

title = {2008, The Airy 1 process is not the limit of the largest eigenvalue in GOE matrix diffusion},

year = {}

}

### OpenURL

### Abstract

Abstract Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1-process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.

### Citations

156 | On orthogonal and symplectic matrix ensembles
- Tracy, Widom
- 1996
(Show Context)
Citation Context ... The stationary distribution of the largest eigenvalue, λN,N(t), can be expressed fairly explicitly, and its limiting distribution, under proper rescaling as N →∞, is the GOE Tracy-Widom distribution =-=[15]-=-. However, no simple expression for the joint distribution of the largest eigenvalue at two different times is known. To be specific, one can ask for the covariance of the largest eigenvalue Cov(λN,N(... |

98 |
A Brownian-motion model for the eigenvalues of a random matrix
- Dyson
- 1962
(Show Context)
Citation Context ...e behavior of the covariance of the Airy1 process, satisfies g1(u) := Cov(A1(u), A1(0)), (2.16) g1(0) = Var(A1(0)) ≃ 0.402 ..., g ′ 1 (0) =−1. (2.17) 3 Dyson’s Brownian Motion Dyson’s Brownian motion =-=[6]-=- describes the diffusion of N mutually repelling particles with positions λj (t), j = 1,...,N, at time t on the real line in a harmonic potential, ( dλj (t) = −γλj (t) + β ∑ 2 i̸=j 1 λj (t) − λi(t) ) ... |

93 | Scale invariance of the PNG droplet and the Airy process
- Prähofer, Spohn
- 2002
(Show Context)
Citation Context ...he limiting process of the properly rescaled largest eigenvalue is the Airy process. This process first arose in the context of one-dimensional stochastic surface growth with curved macroscopic shape =-=[13]-=- for the so-called polynuclear growth (PNG) model. This raised the question, whether the limit process of the largest eigenvalue in GOE matrix diffusion can also be obtained from a growth process. A s... |

76 | Discrete polynuclear growth and determinantal processes
- Johansson
- 2003
(Show Context)
Citation Context ... allows to show the convergence of the rescaled largest eigenvalue, λGUE N,N (u),totheAiry2process, lim N→∞ λGUE N,N (u) = A2(u), (3.6) in the sense of convergence of finite-dimensional distributions =-=[10]-=-. The finite-N covariance of the largest eigenvalue is denoted by f GUE N , f GUE N (u) = Cov ( λ GUE N,N (u), λGUE N,N (0)) . (3.7) Obviously one expects that limN→∞ f GUE N (u) = g2(u). To prove thi... |

29 | Polynuclear growth on a flat substrate and edge scaling of GOE eigenvalues
- Ferrari
(Show Context)
Citation Context ...ne-point distribution is the same as for GOE matrix diffusion [12]. This correspondence was partially extended to a multilayer version of flat growth with non-intersecting height lines. It was shownin=-=[7]-=- that the point process of the multilayer at a fixed position and the point process of the GOE ensemble at the edge of the spectrum have the same asymptotic law. The analogue of the Airy process for f... |

28 |
Universal distributions for growth processes
- Praehofer, Spohn
(Show Context)
Citation Context ...ined from a growth process. A strong candidate was onedimensional growth starting from a flat substrate, since in this case the limiting one-point distribution is the same as for GOE matrix diffusion =-=[12]-=-. This correspondence was partially extended to a multilayer version of flat growth with non-intersecting height lines. It was shownin[7] that the point process of the multilayer at a fixed position a... |

28 |
Spatial correlations of the 1D KPZ surface on a flat substrate
- Sasamoto
- 2005
(Show Context)
Citation Context ...oint process of the GOE ensemble at the edge of the spectrum have the same asymptotic law. The analogue of the Airy process for flat growth was discovered by Sasamoto in a growth model related to PNG =-=[14]-=-. Since its defining kernel at equal times is, in a certain sense, the square root of the standard Airy kernel [9], it was baptized the “Airy1 process”. Accordingly, for better distinction, we call th... |

22 | Fluctuation properties of the TASEP with periodic initial configuration
- Borodin, Ferrari, et al.
(Show Context)
Citation Context ...e square root of the standard Airy kernel [9], it was baptized the “Airy1 process”. Accordingly, for better distinction, we call the standard Airy process “Airy2 process” in the rest of the paper. In =-=[4]-=- two conjectures have been formulated. The first predicted that the Airy1 process is also the limit process for the PNG model with flat initial conditions, which subsequently has beenprovenin[5]. The ... |

21 | P.: PDEs for the joint distributions of the Dyson, Airy and sine processes
- Adler, Moerbeke
- 2005
(Show Context)
Citation Context .... This expression allows to determine some properties of the covariance g2(u) := Cov(A2(u), A2(0)), (2.8) namely g2(0) = Var(A2(0)) = 0.81320 ..., g ′ 2 (0) =−1, (2.9) and the asymptotics for large u =-=[1, 16]-=-, g2(u) = 1 c + u2 u4 + O(u−6 ), (2.10) with the constant c =−3.542 ..., evaluated numerically from an explicit expression in terms of the Hastings-McLeod solution of Painlevé II [2]. 2.3 The Flat PNG... |

19 | Large time asymptotics of growth models on space-like paths I: PushASEP. Available at arXiv.org/abs/0707
- Borodin, Ferrari
- 2007
(Show Context)
Citation Context ...r. In [4] two conjectures have been formulated. The first predicted that the Airy1 process is also the limit process for the PNG model with flat initial conditions, which subsequently has beenprovenin=-=[5]-=-. The second claimed that the Airy1 process is also the limit of the largest eigenvalue in GOE matrix diffusion (β = 1 Dyson’s Brownian motion). In this paper we show that the second conjecture does n... |

17 |
The SIAM 100-Digit Challenge. A
- Bornemann, Laurie, et al.
- 2004
(Show Context)
Citation Context ... exponentially with n, that is, like O(ρ −n ) for some constant ρ > 1. Thus, doubling n doubles the number of correct digits; a fact on which simple strategies for adaptive error control can be based =-=[3]-=-. Additionally, the level of round-off error can be controlled as described in [2]. It turns out that the two-point (m = 2) joint distribution can be calculated to an absolute precision of 10 −14 9us... |

15 | A determinantal formula for the GOE Tracy-Widom distribution
- Ferrari, Spohn
- 2005
(Show Context)
Citation Context ... process for flat growth was discovered by Sasamoto in a growth model related to PNG [14]. Since its defining kernel at equal times is, in a certain sense, the square root of the standard Airy kernel =-=[9]-=-, it was baptized the “Airy1 process”. Accordingly, for better distinction, we call the standard Airy process “Airy2 process” in the rest of the paper. In [4] two conjectures have been formulated. The... |

14 | On asymptotics for the Airy process
- Widom
- 2004
(Show Context)
Citation Context .... This expression allows to determine some properties of the covariance g2(u) := Cov(A2(u), A2(0)), (2.8) namely g2(0) = Var(A2(0)) = 0.81320 ..., g ′ 2 (0) =−1, (2.9) and the asymptotics for large u =-=[1, 16]-=-, g2(u) = 1 c + u2 u4 + O(u−6 ), (2.10) with the constant c =−3.542 ..., evaluated numerically from an explicit expression in terms of the Hastings-McLeod solution of Painlevé II [2]. 2.3 The Flat PNG... |

11 | On the numerical evaluation of Fredholm determinants
- Bornemann
(Show Context)
Citation Context ...or the Airy processes are given in terms of Fredholm determinants of integral operators. To evaluate these Fredholm determinants we employ a numerical scheme, recently developed by one of the authors =-=[2]-=-, which in itself is of general interest. For matrix diffusion we use straightforward Monte-Carlo simulations on large matrices. The comparison shows that the correlation function for GOE matrix diffu... |

8 | The universal Airy1 and Airy2 processes in the Totally Asymmetric Simple Exclusion
- Ferrari
(Show Context)
Citation Context ...) 3) ( 1 − √ exp − 4π(u ′ − u) (s′ − s) 2 4(u ′ ) 1(u − u) ′ >u). (2.15) Some properties of the Airy1 process like the short and long time behavior of the covariance are known. We refer to the review =-=[8]-=- for details. In particular, the short-time behavior of the covariance of the Airy1 process, satisfies g1(u) := Cov(A1(u), A1(0)), (2.16) g1(0) = Var(A1(0)) ≃ 0.402 ..., g ′ 1 (0) =−1. (2.17) 3 Dyson’... |

8 |
symmetric random walks and the extended Hahn kernel
- Johansson, Non-intersecting
(Show Context)
Citation Context ...he weight e−x2 . 3.1 GUE Diffusion and Airy Process In the case β = 2, the point process associated to the ordered eigenvalues, λj (t) of M(t) is determinantal, defined by the extended Hermite kernel =-=[11]-=-. The edge scaling at the upper edge of the spectrum is given by λ GUE N,j (u) = √ 2γN 1/6 ( λj (u/(γ N 1/3 )) − √ ) 2N/γ . (3.5) Under this rescaling, the kernel of the corresponding point process co... |

8 |
private communication
- Widom
- 1996
(Show Context)
Citation Context ...terminant this behavior becomes clear, since one of the off-diagonal blocks is superexponentially small in u for large values of u, while the others stay of order one, a fact already noticed by Widom =-=[17]-=-.414 F. Bornemann et al. Fig. 3 Log-log plot of the rescaled correlation functions for GOE and GUE Finally, in Fig. 3, we provide a comparison of the decay of correlation for GOE and GUE matrix diffu... |

7 |
Waldvogel,“The Siam 100-Digit Challenge: A Study
- Bornemann, Laurie, et al.
- 2004
(Show Context)
Citation Context ...ys exponentially with n, that is, like O(ρ −n ) for some constant ρ>1. Thus, doubling n doubles the number of correct digits; a fact on which simple strategies for adaptive error control can be based =-=[3]-=-. Additionally, the level of round-off error can be controlled as described in [2]. It turns out that the two-point (m = 2) joint distribution can be calculated to an absolute precision of 10 −14 usin... |

3 |
Transition between
- Borodin, Ferrari, et al.
- 2007
(Show Context)
Citation Context ...In [4] two conjectures have been formulated. The first predicted that the Airy1 process is also the limit process for the PNG model with flat initial conditions, which subsequently has been proven in =-=[6]-=-. The second claimed that the Airy1 process is also the limit of the largest eigenvalue in GOE matrix diffusion (β = 1 Dyson’s Brownian motion). In this paper we show that the second conjecture does n... |