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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.05681 |
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| _version_ | 1866915347202834432 |
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| author | Gu, Haotian Zhang, Zhenyuan |
| author_facet | Gu, Haotian Zhang, Zhenyuan |
| contents | We investigate the low moments $\mathbb{E}[|A_N|^{2q}], 0<q\leq 1$ of {secular coefficients} $A_N$ of the {critical non-Gaussian holomorphic multiplicative chaos}, i.e. coefficients of $z^N$ in the power series expansion of $\exp(\sum_{k=1}^\infty X_kz^k/\sqrt{k})$, where $\{X_k\}_{k\geq 1}$ are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper's remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each $X_k$ is standard complex Gaussian, $A_N$ features better-than-square-root cancellation: $\mathbb{E}[|A_N|^2]=1$ and $\mathbb{E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$ for fixed $q\in(0,1)$ as $N\to\infty$. We show that this asymptotics holds universally if $\mathbb{E}[e^{γ|X_k|}]<\infty$ for some $γ>2q$. As a consequence, we establish the universality for the tightness of the normalized secular coefficients $A_N(\log(1+N))^{1/4}$, generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of $\mathbb{E}[|A_N|^{2q}]$ for $|X_k|$ following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper's robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of $A_N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universality and Phase Transitions in Low Moments of Secular Coefficients of Critical Holomorphic Multiplicative Chaos Gu, Haotian Zhang, Zhenyuan Probability We investigate the low moments $\mathbb{E}[|A_N|^{2q}], 0<q\leq 1$ of {secular coefficients} $A_N$ of the {critical non-Gaussian holomorphic multiplicative chaos}, i.e. coefficients of $z^N$ in the power series expansion of $\exp(\sum_{k=1}^\infty X_kz^k/\sqrt{k})$, where $\{X_k\}_{k\geq 1}$ are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper's remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each $X_k$ is standard complex Gaussian, $A_N$ features better-than-square-root cancellation: $\mathbb{E}[|A_N|^2]=1$ and $\mathbb{E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$ for fixed $q\in(0,1)$ as $N\to\infty$. We show that this asymptotics holds universally if $\mathbb{E}[e^{γ|X_k|}]<\infty$ for some $γ>2q$. As a consequence, we establish the universality for the tightness of the normalized secular coefficients $A_N(\log(1+N))^{1/4}$, generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of $\mathbb{E}[|A_N|^{2q}]$ for $|X_k|$ following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper's robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of $A_N$. |
| title | Universality and Phase Transitions in Low Moments of Secular Coefficients of Critical Holomorphic Multiplicative Chaos |
| topic | Probability |
| url | https://arxiv.org/abs/2401.05681 |