Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

Arguin, Louis-Pierre; Belius, David; Bourgade, Paul; Radziwiłł, Maksym; Soundararajan, K.

doi:10.1002/cpa.21791

Cited by 83 publications

(119 citation statements)

References 59 publications

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“…These correspond to replacing N in (118) by log T (c.f [33]). In this case too there has been recent progress in proving the leading order term in the resulting formula when T → ∞ [2,37], based on calculations that mirror those for the extremes of characteristic polynomials. The multiple-integral approach we have developed here also applies to the zeta-function, using the representation established in [12], giving explicit (conjectural) formulae for the integer moments of the integer moments over short intervals of the critical line in that case too.…”

Section: Discussionmentioning

confidence: 99%

“…This is true as well without normalising, in a distributional sense [27]. The correlations of log |P N (A, θ)| can be computed using, for example, formulae due to Diaconis and Shahshahani [16], and shown to satisfy (2) E A∈U (N ) (log |P N (A, θ)| log |P N (A, θ + x)|) ∼ The fact that log |P N (A, θ)| behaves like a log-correlated Gaussian random function has stimulated a good deal of interest recently, as it suggests a connection with other similar random fields such as those associated with the Branching Random Walk, Branching Brownian Motion, the 2-dimensional Gaussian Free Field, and Liouville quantum gravity. This observation, together with heuristic calculations and numerical experiments (c.f.…”

Section: Introductionmentioning

confidence: 99%

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On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Bailey¹,

Keating²

2019

Commun. Math. Phys.

113

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Denoting by PN (A, θ) = det I − Ae −iθ the characteristic polynomial on the unit circle in the complex plane of an N × N random unitary matrix A, we calculate the kth moment, defined with respect to an average over A ∈ U (N ), of the random variable corresponding to the 2βth moment of PN (A, θ) with respect to the uniform measure dθ 2π , for all k, β ∈ N . These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in N of degree k 2 β 2 − k + 1. This resolves a conjecture of Fyodorov & Keating [23] concerning the scaling of the moments with N as N → ∞, for k, β ∈ N. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Bailey¹,

Keating²

2019

Commun. Math. Phys.

113

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“…Arguin et al (2017c); Arguin and Tai (2017); Harper (2013); Saksman and Webb (2016); • The Riemann zeta function on random intervals of the critical line, see e.g. Arguin et al (2017b); Najnudel (2016); • The characteristic polynomials of random unitary matrices, see e.g. Arguin et al (2017a); Chhaibi et al (2017); Paquette and Zeitouni (2016);…”

Section: Motivation For the Scale-inhomogeneous Gffmentioning

confidence: 99%

Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

Ouimet¹

2017

ALEA

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We continue our study of the scale-inhomogeneous Gaussian free field introduced in Arguin and Ouimet (2016). Firstly, we compute the limiting free energy on V N and adapt a technique of Bovier and Kurkova (2004b) to determine the limiting two-overlap distribution. The adaptation was already successfully applied in the simpler case of Arguin and Zindy (2015), where the limiting free energy was computed for the field with two levels (in the center of V N ) and the limiting twooverlap distribution was determined in the homogeneous case. Our results agree with the analogous quantities for the Generalized Random Energy Model (GREM); see Capocaccia et al. (1987) and Bovier and Kurkova (2004a), respectively. Secondly, we show that the extended Ghirlanda-Guerra identities hold exactly in the limit. As a corollary, the limiting array of overlaps is ultrametric and the limiting Gibbs measure has the same law as a Ruelle probability cascade.

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“…Before giving the details, we conclude this section with the following Conjecture. A SLLN as in Theorem 1.1 holds true, mutatis mutandis, in all models belonging to the BBM-universality class, such as the 2-dim Gaussian free field [13,17,9,10,11,12], the 2-dim cover times [20,7,8], the characteristic polynomials of random unitary matrices [1,19,29], and the extreme values of the Riemann zeta function on the critical line [2,28,3,6]. In particular, we expect that an approximate McKean's martingale will capture in all such models the almost sure limit of the normalized number of high-points.…”

Section: Introductionmentioning

confidence: 99%

High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process

Glenz¹,

Kistler²,

Schmidt³

2018

Electron. Commun. Probab.

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It has been proved by Bovier & Hartung [Elect. J. Probab. 19 (2014)] that the maximum of a variable-speed branching Brownian motion (BBM) in the weak correlation regime converges to a randomly shifted Gumbel distribution. The random shift is given by the almost sure limit of McKean's martingale, and captures the early evolution of the system. In the Bovier-Hartung extremal process, McKean's martingale thus plays a role which parallels that of the derivative martingale in the classical BBM. In this note, we provide an alternative interpretation of McKean's martingale in terms of a law of large numbers for high-points of BBM, i.e. particles which lie at a macroscopic distance from the edge. At such scales, 'McKean-like martingales' are naturally expected to arise in all models belonging to the BBM-universality class.

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Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line

Cited by 83 publications

References 59 publications

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Geometry of the Gibbs measure for the discrete 2D Gaussian free field with scale-dependent variance

High points of branching Brownian motion and McKean’s Martingale in the Bovier-Hartung extremal process

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