Secular Coefficients and the Holomorphic Multiplicative Chaos

Abstract

We study the secular coefficients of N × N random unitary matrices U N drawn from the Circular β-Ensemble, which are defined as the coefficients of {z n } in the characteristic polynomial det(1 − zU * N ). When β > 4 we obtain a new class of limiting distributions that arise when both n and N tend to infinity simultaneously. We solve an open problem of Diaconis and Gamburd [DG06] by showing that for β = 2, the middle coefficient of degree n = ⌊ N 2 ⌋ tends to zero as N → ∞. We show how the theory of Gaussian multiplicative chaos (GMC) plays a prominent role in these problems and in the explicit description of the obtained limiting distributions. We extend the remarkable magic square formula of [DG06] for the moments of secular coefficients to all β > 0 and analyse the asymptotic behaviour of the moments. We obtain estimates on the order of magnitude of the secular coefficients for all β > 0, and these estimates are sharp when β ≥ 2. These insights motivated us to introduce a new stochastic object associated with the secular coefficients, which we call Holomorphic Multiplicative Chaos (HMC). Viewing the HMC as a random distribution, we prove a sharp result about its regularity in an appropriate Sobolev space. Our proofs expose and exploit several novel connections with other areas, including random permutations, Tauberian theorems and combinatorics.

A conjectural asymptotic formula for multiplicative chaos in number theory

“…This estimate (and a similar one for all the lower moments E[|A(N )| 2q ] with 0 q 1) was recently established by Soundararajan and Zaman [15] and the upper bound was proved independently by Najnudel, Paquette, and Simm [13]. In view of (1.3), it is natural to conjecture the following asymptotic formula for E[|A(N )|].…”

“…There is a vast literature on each of these topics, so we shall refer the reader to some recent surveys by Rhodes and Vargas [14], Duplantier, Rhodes, Sheffield, and Vargas [7], and Bailey and Keating [1]. On the probability side, Chhaibi and Najnudel [3] and Najnudel, Paquette, and Simm [13] studied the variables A(N ) (which they refer to as "holomorphic multiplicative chaos") to establish direct links between random matrix theory and Gaussian multiplicative chaos. On the number theory side, Soundararajan and Zaman [15] studied A(N ) as a model problem for a breakthrough of Harper [11] on the partial sums of random multiplicative functions.…”

“…Some progress towards the distribution of A(N ) has recently been made by estimating its moments. Notably, building on work of Diaconis and Gamburd [4] with magic squares (see also Gorodetsky [10]), Najnudel, Paquette, and Simm [13] recently proved a beautiful formula: for positive integers q 1, E[|A(N )| 2q ] = #{q × q squares with Z 0 entries and all row and column sums equal to N }. For example, this elegant combinatorial identity implies that the L 2 -moment satisfies…”

Maxima of log-correlated fields: some recent developments

“…Finally, we discuss recent work of Najnudel, Paquette, and Simm [121]. It is now wellestablished that characteristic polynomials can be normalized to converge to GMC measures.…”

“…However, such as statement cannot handle (for example) the middle secular coefficient Sc ⌊ N 2 ⌋ (N). In [121], the authors use Holomorphic multiplicative chaos to show that in the CUE the central secular coefficient tends to zero as N → ∞. Theorem 3.24 (Najnudel, Paquette, Simm [121]).…”