## On the Characteristic Polynomial of a Random Unitary Matrix (2000)

Citations: | 42 - 11 self |

### BibTeX

@MISC{Hughes00onthe,

author = {C. P. Hughes and J.P. Keating and Neil O'connell},

title = {On the Characteristic Polynomial of a Random Unitary Matrix},

year = {2000}

}

### OpenURL

### Abstract

We present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial Z of a random N \Theta N unitary matrix, as N ! 1. First we show that ln Z= ln N , evaluated at a finite set of distinct points, is asymptotically a collection of iid complex normal random variables. This leads to a refinement of a recent central limit theorem due to Keating and Snaith, and also explains the covariance structure of the eigenvalue counting function.

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Citation Context ... ˆ fk = ∫ 2π 0 f(θ)e −ikθ m(dθ) (2.4) its Fourier coefficients. The N th order Toeplitz determinant with symbol f is defined by DN [f ]=det( fj−k)1≤j,k≤N ˆ . (2.5) Heine’s identity (see, for example, =-=[28]-=-) states that DN [f ]=E N∏ f(θn). (2.6) The following lemma is more general than we need here, but we record it for later reference. Lemma 2.3. For any d(N) ≫ 1 as N →∞, s,t ∈ Rk with N sufficiently l... |

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Citation Context .... O’Connell 3. Large Deviations In this section we present large and moderate deviations results for ln Z(0). We begin with a quick review of one-dimensional large deviation theory (see, for example, =-=[8, 11]-=-). We are concerned with the log-asymptotics of the probability distribution of RN /A(N), where RN is some one-dimensional real random variable and A(N) is a scaling that is much greater than the squa... |

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Citation Context ... obtained using Lemma 2.3; however, for many of the results presented here we will need more detailed information. In particular, we will make use of the following explicit formula (see, for example, =-=[2,7, 21]-=-): E exp (sRe ln Z(θ) + tIm ln Z(θ)) G(1 + s/2 + it/2)G(1 + s/2 − it/2)G(1 + N)G(1 + N + s) = , (3.6) G(1 + N + s/2 + it/2)G(1 + N + s/2 − it/2)G(1 + s) valid for Re(s ± it) > −1, where G(·) is the Ba... |

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Citation Context ...again a convex function for which we give an explicit formula. The phase transition reflects a switch from global to local conspiracy. Related fluctuation theorems for random matrices can be found in =-=[9, 12, 11, 19, 14, 26]-=- and references therein. In particular, Diaconis and Evans [11] give an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follo... |

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Citation Context ...group U(N), and denote its eigenvalues by exp(iθ1),... ,exp(iθN ). In order to develop a heuristic understanding of the value distribution and moments of the Riemann zeta function, Keating and Snaith =-=[21]-=- considered the characteristic polynomial (normalised so that its logarithm has zero mean) Z(θ) = det(I − Ue −iθ ) = N∏ ( i(θn−θ) 1 − e ) . (1.1) This is believed to be a good statistical model for th... |

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Citation Context ...again a convex function for which we give an explicit formula. The phase transition reflects a switch from global to local conspiracy. Related fluctuation theorems for random matrices can be found in =-=[9, 12, 11, 19, 14, 26]-=- and references therein. In particular, Diaconis and Evans [11] give an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follo... |

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Citation Context ...th (but do not follow from) a higherlevel large deviation principle due to Hiai and Petz [16]. High-level large deviations results and concentration inequalities for other ensembles can be 3sfound in =-=[4, 5, 15]-=-. Acknowledgements. We are grateful to Persi Diaconis and Steve Evans for their suggestions and for making the preprint [11] available to us. Thanks also to Harold Widom for helpful correspondence. 2 ... |

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Citation Context ...again a convex function for which we give an explicit formula. The phase transition reflects a switch from global to local conspiracy. Related fluctuation theorems for random matrices can be found in =-=[10,13,12,19,14, 27]-=- and references therein. In particular, Diaconis and Evans [12] give an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follo... |

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Citation Context ...ned [3] for all z by G(z + 1) = (2π) z/2 ( exp − 1 ( 2 z 2 + γz 2 )) ∏∞ + z n=1 ( 1 + z n ) n e −z+z2 /2n , (A.1) where γ = 0.5772 ... is Euler’s constant. The G-function has the following properties =-=[3,30]-=-: Recurrence relation: G(z + 1) = Ɣ(z)G(z). Complex conjugation: G∗ (z) = G(z∗ ). Asymptotic formula, valid for |z| →∞with | arg(z)| <π, ln G(z + 1) ∼ z 2 ( 1 2 ln z − 3 ) 1 4 + 2z ln 2π − 1 12 ln z +... |

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Citation Context ...t is more efficient in this case for many eigenvalues to arrange themselves and “share the load”, so to speak, than it is for one to bear it alone. A. Barnes’ G-Function Barnes’ G-function is defined =-=[3]-=- for all z by G(z + 1) = (2π) z/2 ( exp − 1 ( 2 z 2 + γz 2 )) ∏∞ + z n=1 ( 1 + z n ) n e −z+z2 /2n , (A.1) where γ = 0.5772 ... is Euler’s constant. The G-function has the following properties [3,30]:... |

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Citation Context ...tral limit theorem with a central limit theorem, due to Selberg, for the value distribution of the log of the Riemann zeta function along the critical line. Selberg proved (see, for example, §2.11 of =-=[24]-=- or §4 of [22]) that, for rectangles B ⊆ C, 1 lim T →∞ T ⎧ ⎨ ln ζ( T ≤ t ≤ 2T : ∣⎩ 1 ⎫ 2 + it) ⎬ √ ∈ B 12 ⎭ ln ln T ∣ ∫∫ 1 = e 2π B −(x2 +y2)/2 dx dy. (1.5) Equating the mean density of the Riemann ze... |

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Citation Context .... complex normal random variables. This leads to a refinement of the above central limit theorem, and also explains the mysterious covariance structure which has been observed, by Costin and Lebowitz =-=[9]-=- and Wieand [32, 33], in the eigenvalue counting function. We also obtain a central limit theorem for In Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lowe... |

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Citation Context ... (2.30) k=1 ∞∑ k 2b−1 ; (2.31) k=1436 C. P. Hughes, J. P. Keating, N. O’Connell a similar bound holds for Im ln Z. We have used the fact that E| Tr U k | 2 = min(|k|,N) for k = 0 (see, for example, =-=[25]-=-). Thus, sup P (max{‖Re ln Z‖b, ‖Im ln Z‖b} >q) (2.32) N ≤ sup {P (‖Re ln Z‖b >q)+ P (‖Im ln Z‖b >q)} (2.33) N { E‖Re ln Z‖ 2 b + E‖Im ln Z‖2 } b /q 2 ≤ sup (2.34) N → 0 (2.35) as q →∞, so we are done... |

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Citation Context ...l random variables. This leads to a refinement of the above central limit theorem, and also explains the mysterious covariance structure which has been observed, by Costin and Lebowitz [9] and Wieand =-=[32, 33]-=-, in the eigenvalue counting function. We also obtain a central limit theorem for In Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which ... |

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Citation Context ...an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follow from) a higherlevel large deviation principle due to Hiai and Petz =-=[16]-=-. High-level large deviations results and concentration inequalities for other ensembles can be 3sfound in [4, 5, 15]. Acknowledgements. We are grateful to Persi Diaconis and Steve Evans for their sug... |

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Citation Context ... Lemma 2.3 For any d(N) » 1 as N ---+ 00, s, t E IRk with N sufficiently 5slarge such that Sj > -d(N) for all j, and rj distinct in 1', Proof. This follows from Heine's identity and a result of Basor =-=[3]-=- on the asymptotic behaviour of Toeplitz determinants with Fisher-Hartwig symbols. The Fisher-Hartwig symbol we require has the form k f(()) = II(1 - e i (8-rj) t j +,8j (1 - e i (r j -8)t j -,8j. (2.... |

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Citation Context ... random variables. This leads to a refinement of the above central limit theorem, and also explains the mysterious covariance structure which has been observed, by Costin and Lebowitz [10] and Wieand =-=[32,33]-=-, in the eigenvalue counting function. We also obtain a central limit theorem for ln Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which ... |

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Citation Context ...Z(0) √ �⇒ X + iY, (1.2) 12 ln N where X and Y are independent normal random variables with mean zero and variance one 1 , and �⇒ denotes convergence in distribution. (A similar result can be found in =-=[2]-=-, but there the real and imaginary parts of ln Z/σ are treated separately.) In order to make the imaginary part of the logarithm well-defined, the branch is chosen so that and ln Z(θ) = N∑ n=1 ( ) i(θ... |

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Citation Context ...ith (but do not follow from) a higher-level large deviation principle due to Hiai and Petz [16]. High-level large deviations results and concentration inequalities for other ensembles can be found in =-=[5,6,15]-=-. 2. Fluctuation Results Our first main result is that the law of ln Z(0) obtained by averaging over the unitary group is asymptotically the same as the value distribution of ln Z(θ) obtained by avera... |

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Citation Context ... n=1430 C. P. Hughes, J. P. Keating, N. O’Connell (1/2π)ln(T /2π)) is set equal to the mean density of eigenangles (which is N/2π). (For additional evidence of this, concerning other statistics, see =-=[9]-=-.) Note that the law of Z(θ) is independent of θ ∈ T (the unit circle). In [21] it is shown that as N →∞,lnZ(0)/σ converges in distribution to a standard complex normal random variable, where 2σ 2 = l... |

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Citation Context ... follow from Theorem 2.2 below, so we defer the proof. Theorem 2.1 hints at the possibility that the range in (1.4) can be significantly reduced, even beyond the refinements already obtained by Ghosh =-=[13]-=- (see also [21D. The characteristic polynomial can also be used to explain the mysterious 'white noise' process which appears in recent work of Wieand [32, 33] on the counting function (and less expli... |

8 |
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Citation Context ...verges in distribution to a collection of i.i.d. standard complex normal random variables. (In fact, it is shown in [13] that there is exact agreement of moments up to high order for each N. See also =-=[18]-=-, where superexponential rates of convergence are established.) Denote by H a 0 the space of generalised real-valued functions f on T with f0 ˆ = 0 and ‖f ‖ 2 a = ∞∑ k=−∞ |k| 2a | ˆ fk| 2 = 2 This is ... |

8 |
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Citation Context ...i(rm -rn)) -(Om-,8m)(On+,8n) Fal,{31,rl, ... ,ak, k,rk - 1-e X j=1 m:;i:n k XII G(l + aj + (3j)G(l + aj - (3j) j=1 G(l + 2aj) 6swhere larg (1- ei(rm-r n »)I::; 7r/2. By closer inspection of the proof =-=[31]-=- it can be seen that (2.8) holds uniformly for lail < 1/2 - 8, and l,8il < "f, for any fixed 8, "f > 0. This is worked out carefully in [32] in the case ai =°for each j, and uniformity in a is discuss... |

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Citation Context ...C → R defined by f(z) = h(arg z)δ(|z| =1) for some h : T → R with ˆh0 = 0. Then, as t →∞, 1 π √ ln t ∫ t 0 f(Bs)ds �⇒ 〈h, F 〉. (2.49) This can be deduced from a result of Kasahara and Kotani given in =-=[20]-=-.438 C. P. Hughes, J. P. Keating, N. O’Connell 3. Large Deviations In this section we present large and moderate deviations results for ln Z(0). We begin with a quick review of one-dimensional large ... |

4 |
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Citation Context ...,' .• ,0,0, Tk) = 1. 0 Setting d = a completes the proof of Theorem 2.2. 0 Proofof Theorem 2.1. Set X N ((}) = ryteln Z((})/a, YN ((}) = Jmln Z((})/a and (2.9) By the central limit theorem derived in =-=[20]-=- (we note, in passsing, that this also follows from Theorem 2.2), We also have By Cauchy-Schwartz, the integrand is bounded above by sup Eexp (2sXN (0) + 2tYN (0)) , N?No (2.12) where No is chosen suc... |

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Citation Context ...th (but do not follow from) a higherlevel large deviation principle due to Hiai and Petz [16]. High-level large deviations results and concentration inequalities for other ensembles can be 3sfound in =-=[4, 5, 15]-=-. Acknowledgements. We are grateful to Persi Diaconis and Steve Evans for their suggestions and for making the preprint [11] available to us. Thanks also to Harold Widom for helpful correspondence. 2 ... |

1 |
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Citation Context ...again a convex function for which we give an explicit formula. The phase transition reflects a switch from global to local conspiracy. Related fluctuation theorems for random matrices can be found in =-=[9, 12, 11, 19, 14, 26]-=- and references therein. In particular, Diaconis and Evans [11] give an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follo... |

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Citation Context ...l random variables. This leads to a refinement of the above central limit theorem, and also explains the mysterious covariance structure which has been observed, by Costin and Lebowitz [9] and Wieand =-=[32, 33]-=-, in the eigenvalue counting function. We also obtain a central limit theorem for In Z in a Sobolev space of generalised functions on the unit circle. In this limiting regime, lower-order terms which ... |

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Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields
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Citation Context ...again a convex function for which we give an explicit formula. The phase transition reflects a switch from global to local conspiracy. Related fluctuation theorems for random matrices can be found in =-=[10,13,12,19,14, 27]-=- and references therein. In particular, Diaconis and Evans [12] give an alternative proof of Theorem 2.2 below. The large deviation results at speed N 2 are partially consistent with (but do not follo... |