Large deviations and continuity estimates for the derivative of a random model of log⁡|ζ| on the critical line

Abstract

In this short note, we study the derivatives of all orders for the random fieldwhere (U p , p primes) is an i.i.d. sequence of uniform random variables on the unit circle in C. We show that the maximum of X T , and more generally its j-th derivative, varies on a (log T ) − 1 2 (j+2) scale, which improves and extends the main result in Arguin & Ouimet (2019) and makes further progress towards the open problem of the tightness of the recentered maximum of X T . Our proof is also much simpler and shorter.

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

“…uniform random variables on the unit circle, cf. Harper (2013b); Arguin, Belius and Harper (2017); Arguin and Ouimet (2019). The analogue of conjecture (1.6) for this model was proved up to second-order corrections in Arguin, Belius and Harper (2017), and large deviations and continuity estimates for the derivative were found in Arguin and Ouimet (2019).…”

“…Harper (2013b); Arguin, Belius and Harper (2017); Arguin and Ouimet (2019). The analogue of conjecture (1.6) for this model was proved up to second-order corrections in Arguin, Belius and Harper (2017), and large deviations and continuity estimates for the derivative were found in Arguin and Ouimet (2019). The limit of the corresponding multiplicative chaos measure was obtained in Saksman and Webb (2018).…”

“…Proof. For (A.1), see Lemma A.1 of Arguin and Ouimet (2019) and Lemma 2.1 of Arguin, Belius and Harper (2017). For (A.2), see p.20 in Harper (2013b).…”

Maxima of branching random walks with piecewise constant variance

“…• cover times (see, e.g., Abe (2014Abe ( , 2018, Belius (2013), Belius and Kistler (2017), Comets et al (2013), Dembo, Peres and Rosen (2003), Dembo et al (2004Dembo et al ( , 2006, Ding (2012Ding ( , 2014, Ding, Lee and Peres (2012), Ding and Zeitouni (2012)); • the extremes of the randomized Riemann zeta function on the critical line (see, e.g., Arguin and Ouimet (2018), Arguin and Tai (2018), Arguin, Belius and Harper (2017), Harper (2013), Ouimet (2018), Saksman and Webb (2018) show that approximate branching structures are present in a huge variety of models. Hence, the approach of this paper might become relevant in applications beyond the study of "pure" BRW.…”

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

“…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]). 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).…”