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Haotian Gu : Universality and Phase Transitions of Holomorphic Multiplicative Chaos

The random distribution Holomorphic multiplicative chaos (HMC) with Gaussian inputs is recently introduced independently by Najnudel, Paquette, and Simm as a limiting object on the unit complex circle of characteristic polynomial of circular beta ensembles, and by Soundararajan and Zaman as an analogue of random multiplicative functions. In this talk, we will explore this rich connection between HMC and random matrix theory, number theory, and Gaussian multiplicative chaos. We will also discuss the regularity of this distribution, alongside the fractional moments and tightness of its Fourier coefficients (also referred to as secular coefficients). Furthermore, we introduce non-Gaussian HMC, and discuss the Gaussian universality and two phase transitions phenomenon in the fractional moments of its secular coefficients. A transition from global to local effect is observed, alongside an analysis of the critical local-global case. As a result, we unveil the regularity of some non-Gaussian HMC and tightness of their secular coefficients. Based on joint work with Zhenyuan Zhang.

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