On Gaussian multiplicative chaos

Gaussian multiplicative chaos was first constructed in Kahane's seminal paper in 1985 in an attempt to provide a mathematical foundation for Kolmogorov-Obukhov-Mandelbrot theory of energy dissipation in developed turbulence. It has attracted a lot of attentions from the mathematics community in the last decade, playing a pivotal role in the probabilistic formulation of Liouville conformal field theory, as well as showing up in different branches of mathematics such as analytic number theory where it describes the statistical behaviour of the Riemann zeta function on the critical line. This thesis explores the theory of Gaussian multiplicative chaos in three different directions. We commence with a new connection with random matrix theory, showing that for large Hermitian matrices sampled from the one-cut-regular unitary ensemble, the absolute powers of the characteristic polynomial, when suitably normalised, converge in distribution to multiplicative chaos on the support of the limiting spectral distribution as the size of the matrix goes to infinity, and the limit is independent of the choice of the potential function. This is part of an ongoing programme of establishing Gaussian multiplicative chaos as a universal limit object in probability theory. Next, we consider Gaussian multiplicative chaos in the context of Liouville conformal field theory and study the fusion estimate of the Liouville correlation function. More precisely, we derive the exact asymptotics for the Liouville four-point correlation when two points are merging and express the leading order coefficient in terms of DOZZ constants from the three-point correlation function. Our result is consistent with predictions from conformal bootstrap in theoretical physics, and has a geometric interpretation of surfaces being glued together, as hinted by the bootstrap equation. Finally, we study the right tail of the mass of Gaussian multiplicative chaos and establish a formula for the leading order asymptotics under mild assumptions on the underlying log-correlated Gaussian field. The tail exponent satisfies a universal power-law profile, while the leading order coefficient can be described by the product of two constants, one capturing the dependence on the test set and any non-stationarity, and the other one encoding the universal properties of multiplicative chaos. This may be seen as a first step in understanding the full distributional properties of Gaussian multiplicative chaos.