Is the Riemann Zeta Function in a Short Interval a 1-RSB Spin Glass?

Arguin, Louis-Pierre; Tai, Warren

doi:10.1007/978-981-15-0294-1_3

Cited by 5 publications

(12 citation statements)

References 34 publications

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“…In Section 3, the main result is stated and shown to be a consequence of the GG identities and the main result from [4] about the limiting two-overlap distribution. In Section 4, we state known results from [4] that we will use to prove the GG identities. The GG identities are proven in Section 5 along with other preliminary results, see the structure of the proof in Figure 5.1.…”

Section: Introductionmentioning

confidence: 96%

“…is a sequence of random variables converging in distribution. For a randomized version of the Riemann zeta function (see (2.1)), the first order of the maximum was proved in [16], the second order of the maximum was proved in [2], and the limiting two-overlap distribution was found in [4] (see Theorem 3.1 below). The tightness of the recentered maximum is still open (see [3]).…”

Section: Introductionmentioning

confidence: 99%

“…The tightness of the recentered maximum is still open (see [3]). In this short paper, we complete the analysis of [4] by proving the Ghirlanda-Guerra (GG) identities in the limit T → ∞ (see Theorem 5.8). As is well known in the spin glass literature (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Ouimet¹

2018

Electron. Commun. Probab.

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In [4], the authors prove the convergence of the two-overlap distribution at low temperature for a randomized Riemann zeta function on the critical line. We extend their results to prove the Ghirlanda-Guerra identities. As a consequence, we find the joint law of the overlaps under the limiting mean Gibbs measure in terms of Poisson-Dirichlet variables. It is expected that we can adapt the approach to prove the same result for the Riemann zeta function itself.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Ouimet¹

2018

Electron. Commun. Probab.

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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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Prime zeta function statistics and Riemann zero-difference repulsion

Chavez¹,

Allawala²

2021

J. Stat. Mech.

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We present a derivation of the numerical phenomenon in which differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical 'repulsion' between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the one-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.

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Is the Riemann Zeta Function in a Short Interval a 1-RSB Spin Glass?

Cited by 5 publications

References 34 publications

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

Poisson-Dirichlet statistics for the extremes of a randomized Riemann zeta function

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

Prime zeta function statistics and Riemann zero-difference repulsion

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