Subcritical Multiplicative Chaos for Regularized Counting Statistics from Random Matrix Theory

Abstract

For an N × N Haar distributed random unitary matrix UN , we consider the random field defined by counting the number of eigenvalues of UN in a mesoscopic arc centered at the point u on the unit circle. We prove that after regularizing at a small scale ǫN > 0, the renormalized exponential of this field converges as N → ∞ to a Gaussian multiplicative chaos measure in the whole subcritical phase. We discuss implications of this result for obtaining a lower bound on the maximum of the field. We also show that the moments of the total mass converge to a Selberg-like integral and by taking a further limit as the size of the arc diverges, we establish part of the conjectures in [55]. By an analogous construction, we prove that the multiplicative chaos measure coming from the sine process has the same distribution, which strongly suggests that this limiting object should be universal. Our approach to the L 1 -phase is based on a generalization of the construction in Berestycki [5] to random fields which are only asymptotically Gaussian. In particular, our method could have applications to other random fields coming from either random matrix theory or a different context.

On the moments of the characteristic polynomial of a Ginibre random matrix

“…What one would expect from these results is that converges in law to a multiplicative chaos measure as . Moreover, a central question in is to have precise asymptotics for quantities corresponding to , so Theorem is a first step in this direction as well.…”

“…Thus, motivated by the central limit theorem of Rider and Virág, a natural question is whether multiplicative chaos measures can be constructed from the characteristic polynomials of the Ginibre ensemble and can the limiting measure be connected to these objects appearing in random geometry. Recently, multiplicative chaos measures have been constructed from characteristic polynomials of random matrices in the setting of random unitary and random Hermitian matrices — see . What one would expect from these results is that converges in law to a multiplicative chaos measure as .…”

“…converges in law to a multiplicative chaos measure as N → ∞. Moreover, a central question in [6,29,37] is to have precise asymptotics for quantities corresponding to E k j=1 | det(G N − z j )| γj , so Theorem 1.1 is a first step in this direction as well.…”

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

“…[31], [39], upon which GMC measures are built, from the complexity of mathematical problems that their stochastic dependence poses, and from the many applications in mathematical and theoretical physics and pure mathematics, in which GMC naturally appears. Without aiming for comprehension, we can mention applications to five areas: (1) conformal field theory and Liouville quantum gravity [3], [11], [29], [83], [85], (2) statistical mechanics of disordered energy landscapes [18], [19], [22], [37], [38], [39], [42], [43], [78], (3) random matrix theory [45], [46], [48], [57], [98], (4) statistical modeling of fully intermittent turbulence [23], [24], [25], [35], [68], [84], (5) conjectured [40], [44], [77] and some rigorous [90] applications to the behavior of the Riemann zeta function on the critical line.…”

Universal tail profile of Gaussian multiplicative chaos

“…First introduced by Kahane [21] in an attempt to provide a mathematical framework for Kolmogorov-Obukhov-Mandelbrot's model of turbulence, the theory of GMCs has attracted a lot of attention in the probability and mathematical physics community in the last decade due to its central role in random planar geometry [16,19] and Liouville conformal field theory [14], and new applications in e.g. random matrix theory [35,26,11,27,13]. Various equivalent constructions of Mγ have been studied, including the mollification approach which proceeds by considering the weak * limit of measures…”