论文标题
第三阶段中复杂高斯乘法混乱的法律收敛
Convergence in law for Complex Gaussian Multiplicative Chaos in phase III
论文作者
论文摘要
高斯乘法混乱（GMC）被非正式地定义为一个随机度量$ e^{γx} \ mathrm {d} x $其中$ x $是$ \ mathbb r^d $上的高斯字段（或它的开放子集）的相关功能是$ k（x，y） l（x，y），$ l $是连续函数$ x $和$ y $，$γ=α+iβ$是一个复杂的参数。在本文中，我们考虑了\ Mathcal p'_ {\ Mathrm {iii}} $中的情况，其中$ \ Mathcal p'_ {\ Mathrm {\ Mathrm {iii}}：= \ = \ {α+iβ\：α+iβ\：α，γ\：α，γ\ in \ in \ in \ mathbb r，c \ s < \ α^2+β^2\ge d \}.$$ We prove that if $X$ is replaced by the approximation $X_\varepsilon$ obtained by convolution with a smooth kernel, then $e^{γX_\varepsilon} \mathrm d x$, when properly rescaled, has an explicit non-trivial limit in distribution when $ \ varepsilon $变为零。该极限不取决于用于定义$ x _ {\ varepsilon} $的特定卷积内核，并且可以描述为复杂的高斯白噪声，其随机强度由与参数$2α$相关的真实GMC给出。
Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure $e^{γX} \mathrm{d} x$ where $X$ is Gaussian field on $\mathbb R^d$ (or an open subset of it) whose correlation function is of the form $ K(x,y)= \log \frac{1}{|y-x|}+ L(x,y),$ where $L$ is a continuous function $x$ and $y$ and $γ=α+iβ$ is a complex parameter. In the present paper, we consider the case $γ\in \mathcal P'_{\mathrm{III}}$ where $$ \mathcal P'_{\mathrm{III}}:= \{ α+i β\ : α,γ\in \mathbb R , \ |α|<\sqrt{d/2}, \ α^2+β^2\ge d \}.$$ We prove that if $X$ is replaced by the approximation $X_\varepsilon$ obtained by convolution with a smooth kernel, then $e^{γX_\varepsilon} \mathrm d x$, when properly rescaled, has an explicit non-trivial limit in distribution when $\varepsilon$ goes to zero. This limit does not depend on the specific convolution kernel which is used to define $X_{\varepsilon}$ and can be described as a complex Gaussian white noise with a random intensity given by a real GMC associated with parameter $2α$.
