Basic properties of critical lognormal multiplicative chaos

Abstract

We study one-dimensional exact scaling lognormal multiplicative chaos measures at criticality. Our main results are the determination of the exact asymptotics of the right tail of the distribution of the total mass of the measure, and an almost sure upper bound for the modulus of continuity of the cumulative distribution function of the measure. We also find an almost sure lower bound for the increments of the measure almost everywhere with respect to the measure itself, strong enough to show that the measure is supported on a set of Hausdorff dimension 0.

Moments of Random Multiplicative Functions, I: Low Moments, Better Than Squareroot Cancellation, and Critical Multiplicative Chaos

“…EX(jE⌊log log x⌋) 2 ≥ u , 3 Later we will also need to know that the covariance is a decreasing function of |j − k|, on a suitable range. To do this, we note that if we differentiate, where u = u(β, x) is given by u := (1/2) log β + log log x EX(jE⌊log log x⌋) 2 = (1/2) log β + log log x (1/2)(log log x − 2 log log log x) + O(1) ≥ 2 log log x.…”

Universal tail profile of Gaussian multiplicative chaos

“…(3.1) Lemma 3.1 is essentially due to [5], where the d ≤ 2 cases were established as Theorem 1 and Theorem 25 there. Quoting the discussion before the Appendices in [5], the proof of Lemma 3.1 for d ≤ 2 may be extended to higher dimensions immediately as long as one has the existence of the corresponding critical chaos (which has now been addressed) and an estimate analogous of [5, Lemma 29] for d ≥ 3. For later applications we state and prove this analogous result for general GMCs.…”

“…The proof of Lemma 3.2 is postponed to Appendix A, and we refer the readers to [5] for the arguments leading to a proof of Lemma 3.1.…”

“…However, Lemma 3.2 suggests the existence of some h > 1 2 such that The splitting lemma, while simple and based on Lemma 3.2, was not known in [5] and its power could not have been harnessed. For an illustration, we have Corollary 3.6.…”

On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral

“…Theorem 2.1 applies to f (i) ε,u (x) for all u > ε > 0. Fix ε > 0 and denote the BKR statistic based on some constant 8…”