A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

Abstract

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of 1/ f noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model.

The distribution of Gaussian multiplicative chaos on the unit interval

“…Thus from this relation and knowing that M (γ, 0, a, b) = 1 one can compute recursively all the negative moments of the random variable I γ,a,b . As it was emphasized in many papers (see the review [25] by Ostrovsky and references therein), the negative moments of I γ,a,b do not determine its law as the growth of the negative moments is too fast. This is why we must derive a second relation between M (γ, p, a, b) and M (γ, p − 4 γ 2 , a, b) which gives enough information to complete the proof.…”

“…This function is useful when we study the limit γ → 2 in section 1.4. Finally in our Corollary 1.3 we have used a special β 2,2 distribution defined in [25]. Here we recall the definition: 7 In [25] Ostrovsky uses a slightly different special function Γ2(x|τ ), the relation with our Γ γ 2 (x) is:…”

“…However a crucial analycity argument is missing for this approach to prove rigorously an exact formula. See [25] for a beautiful review on all the known results and conjectures for the GMC on the interval (and also for the similar model on the circle) as well as for many additional references.…”

Statistics of extremes in eigenvalue-counting staircases

“…Note the non trivial term 2 log(β/2) in the last expression, arising from the replacement −2 log r = −2 log s+2 log(β/2) in (43). For β = 2 one recovers C(β=2)…”

“…Through exact solutions, it led to predictions for the PDF of the maximum value of a log-correlated field on the circle and on the interval [40,41], involving the so-called freezing duality conjec-ture (FDC) (see [21] for an extensive discussion of this conjecture). This led to intensive studies and further results in theoretical and mathematical physics [24,[42][43][44][45][46] and probability [25,[47][48][49][50][51][52][53][54][55]. One should remember that log-correlated fields are essentially random generalised functions (distributions) and when discussing their value distribution and extrema one necessarily works with regularizations of such highly singular object.…”