Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem

Amzallag, Eli; Arguin, Louis-Pierre; Bailey, Emma; Huib, K.; Rao, R.

doi:10.1080/10586458.2021.2011806

Cited by 4 publications

(6 citation statements)

References 21 publications

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“…In [103], the authors provide numerical evidence in support of (204). The parameter m in (207) contains two constants of interest.…”

Section: Theorem 213 (Arguin Et Al [133]) Take θ > −1 Let τ Be Unifor...mentioning

confidence: 63%

“…Figure 6 shows plots of log |ζ(1/2 + it)| for t in short ranges near 5000 000. Since the ranges of ( 93) and (95) are considerably shorter than the O(T) lengths considered in ( 111)- (114), experimental calculations are also feasible [102,103].…”

Section: Theorem 21 (Diaconis and Shashahani [92]mentioning

confidence: 99%

“…Secondly, f k is the leading order coefficient of the 2kth moment of ζ (43). The appearance of the moment coefficient is related to large deviations of Selberg's central limit theorem [103,136,137]. Indeed, one can show 35 that the equivalent moment parameter appears in the large deviations of the Keating and Snaith central limit theorem 1.3.…”

Section: Theorem 213 (Arguin Et Al [133]) Take θ > −1 Let τ Be Unifor...mentioning

confidence: 99%

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Maxima of log-correlated fields: some recent developments*

Bailey

Keating²

2022

J. Phys. A: Math. Theor.

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We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.

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“…In [103], the authors provide numerical evidence in support of (204). The parameter m in (207) contains two constants of interest.…”

Section: Theorem 213 (Arguin Et Al [133]) Take θ > −1 Let τ Be Unifor...mentioning

confidence: 63%

Section: Theorem 21 (Diaconis and Shashahani [92]mentioning

confidence: 99%

Section: Theorem 213 (Arguin Et Al [133]) Take θ > −1 Let τ Be Unifor...mentioning

confidence: 99%

See 1 more Smart Citation

Maxima of log-correlated fields: some recent developments*

Bailey

Keating²

2022

J. Phys. A: Math. Theor.

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“…The resulting model (1.7), when exponentiated, can thus be seen as a random Euler product. 1 It was shown in [3] that with high probability…”

Section: Resultsmentioning

confidence: 99%

“…Indeed, one expects non-trivial arithmetic and random-matrix-type corrections of order one whenever θ > 0. Numerical evidence of this phenomenon will appear in [1].…”

Section: Structure Of the Proofmentioning

confidence: 93%

Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length

Arguin¹,

Dubach²,

Hartung³

2021

Preprint

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We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length (log T ) θ , where θ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from 3 4 to 1 4 ' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is a precise upper tail tightness estimate for the maximum of the model on intervals of size one, that includes a Gaussian correction. This correction is expected to be present for the Riemann zeta function and pertains to the question of the correct order of the maximum of the zeta function in large intervals.

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Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications

Arguin

Bailey

2023

International Mathematics Research Notices

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For $V\sim \alpha \log \log T$ with $0<\alpha <2$, we prove $$\begin{align*} & \frac{1}{T}\textrm{meas}\{t\in [T,2T]: \log|\zeta(1/2+ \textrm{i} t)|>V\}\ll \frac{1}{\sqrt{\log\log T}} e^{-V^{2}/\log\log T}. \end{align*}$$This improves prior results of Soundararajan and of Harper on the large deviations of Selberg’s Central Limit Theorem in that range, without the use of the Riemann hypothesis. The result implies the sharp upper bound for the fractional moments of the Riemann zeta function proved by Heap, Radziwiłł, and Soundararajan. It also shows a new upper bound for the maximum of the zeta function on short intervals of length $(\log T)^{\theta }$, $0<\theta <3$, that is expected to be sharp for $\theta> 0$. Finally, it yields a sharp upper bound (to order one) for the moments on short intervals, below and above the freezing transition. The proof is an adaptation of the recursive scheme introduced by Bourgade, Radziwiłł, and one of the authors to prove fine asymptotics for the maximum on intervals of length $1$.

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Evidence of Random Matrix Corrections for the Large Deviations of Selberg’s Central Limit Theorem

Cited by 4 publications

References 21 publications

Maxima of log-correlated fields: some recent developments*

Maxima of log-correlated fields: some recent developments*

Maxima of a Random Model of the Riemann Zeta Function over Intervals of Varying Length

Large Deviation Estimates of Selberg’s Central Limit Theorem and Applications

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