Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convergence in law for Complex Gaussian Multiplicative Chaos in phase
  III   Gaussian Multiplicative Chaos (GMC) is informally defined as a random measure
$e^{\gamma X} \mathrm{d} x$ 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
$\gamma=\alpha+i\beta$ is a complex parameter. In the present paper, we
consider the case $\gamma\in \mathcal P'_{\mathrm{III}}$ where $$ \mathcal
P'_{\mathrm{III}}:= \{ \alpha+i \beta \ : \alpha,\gamma \in \mathbb R , \
|\alpha|<\sqrt{d/2}, \ \alpha^2+\beta^2\ge d \}.$$ We prove that if $X$ is
replaced by the approximation $X_\varepsilon$ obtained by convolution with a
smooth kernel, then $e^{\gamma X_\varepsilon} \mathrm d x$, 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 $2\alpha$.

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The categories of this paper are: econ
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
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The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: stat
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The categories of this paper are: comp-gas