| Russian Mathematical Surveys |
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Brief communications Gaussian multiplicative chaos for the sine-process A. I. Bufetovabcd a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow b Institute for Information Transmission Problems of the Russian Academy of Sciences (Kharkevich Institute) c Saint Petersburg State University d CNRS, Institut de Mathématiques de Marseille, Marseille, France
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 6(474), DOI: https://doi.org/10.4213/rm10156 
Bibliographic databases:
    Let the symbol $\mathsf{P}_{\mathcal S}$ denote the sine-process, the determinantal measure on the space $\operatorname{Conf}(\mathbb{R})$ of locally finite configurations on $\mathbb{R}$ that is induced by the kernel For any $z,w\in\mathbb{C}\setminus\mathbb{R}$ we introduce a multiplicative functional by the formula
$$
\begin{equation*}
G_X(z,w)=\prod_{x\in X}^{\mathrm{v.p.}}\biggl(\frac{z-x}{w-x}\biggr),\qquad x\in X.
\end{equation*}
\notag
$$
Furthermore, let $M_\gamma$ be the exponential of the Gaussian random field on $[0,1]$ with covariance kernel $K_\gamma(s,t)=-2\gamma^2\log|s-t|$ (see [7], [9], and [10]).
Theorem. For $\gamma\in(0,1)$ the random measure converges in distribution to $M_{\gamma}$ as $R\to\infty$. In other words, the distribution of the random measure (1) in the space of probability measures on the space of finite Borel measures on $[0, 1]$ converges weakly to the distribution of the random measure $M_{\gamma}$. Let $\mathcal{PW}=\{f\in L_2(\mathbb{R})\colon \operatorname{supp}\hat f\subset [-\pi,\pi]\}$. Corollary. For $\mathsf{P}_{\mathcal S}$-almost every configuration $X\in\operatorname{Conf}(\mathbb{R})$ and any $p\in X$ the conditions $f\in\mathcal{PW}$ and $f\big|_{X\setminus p}=0$ imply that $f=0$. The set $X\setminus p$ is therefore complete for the Paley–Wiener space. The completeness of $X$ itself was established by Gosh [5] (see also [8] for the discrete case and [3] for the general case). On the other hand the following statement holds. Proposition. For $\mathsf{P}_{\mathcal S}$-almost every configuration $X\in\operatorname{Conf}(\mathbb{R})$ and any distinct $p,q\in X$ there exists a non-trivial function $f\in\mathcal{PW}$ such that $f\big|_{X\setminus\{p,q\}}=0$. We conclude that the set $X\setminus p$ is both complete and minimal for the Paley-Wiener space $\mathcal{PW}$. Consider a Gaussian random process $Y_{zw}$, indexed by points $z$ and $w$ in the upper half-plane $\mathbb{C}_+=\{z\in\mathbb{C}\colon\operatorname{Im} z>0\}$, that is specified by the following conditions:
$$
\begin{equation*}
\begin{gathered} \, Y_{zw}+Y_{wu}+Y_{uz}=0,\quad \mathsf{E}Y_{zw}=0,\\ \text{and}\quad \operatorname{Var}Y_{zw}=\log\biggl(1+\frac{|z-w|^2}{4\operatorname{Im} z\operatorname{Im} w}\biggr),\quad u,z,w\in\mathbb{C}_+. \end{gathered}
\end{equation*}
\notag
$$
By the Soshnikov central limit theorem [11] (see also [6]) the random variables $\log|G_X(z/R,w/R)|$ converge in distribution to the Gaussian random process $Y_{zw}$ as $R\to\infty$. This convergence is uniform for $z$ and $ w$ ranging over compact subsets of the upper half-plane. It is natural to consider $Y_{zw}$ as a process indexed by pairs of points in the Lobachevskii plane. Indeed, the joint distributions of the random variables we have introduced remain invariant under the group of Lobachevskian isometries. If $z$ is fixed and $w$ approaches the absolute, then the random process $Y_{zw}$ converges to the Gaussian random field with logarithmic correlations. In order to prove the convergence of the random measures (1) we employ the scaling limit of the Borodin–Okounkov–Geronimo–Case formula [1], [2], [4] that we now formulate. Take any $f\in L_\infty(\mathbb R)$ satisfying the following conditions
$$
\begin{equation*}
\|f\|^2_{\dot H_{1/2}}=\iint_{\mathbb{R}^2} \biggl|\frac{f(x)-f(y)}{x-y}\biggr|^2\,dx\,dy<\infty\quad\text{and}\quad \mathrm{(v.p.)}\int_{-\infty}^{+\infty}f(x)\,dx=0.
\end{equation*}
\notag
$$
Represent $f$ as a sum $f=f_++f_-$, where $\operatorname{supp} \widehat{f_+}\subset [0,+\infty)$ and $\operatorname{supp} \widehat{f_-}\subset (-\infty,0]$. Then
$$
\begin{equation*}
\det(1+(e^f-1){\mathcal S})=\exp\biggl\{ \frac{1}{4\pi^2}\|f\|^2_{\dot H_{1/2}}\biggr\}\cdot \det\bigl(1-\mathbf{1}_{(1,+\infty)}\mathfrak{H}(h_{-+}) \mathfrak{H}(\widetilde{h_{+-}})\mathbf{1}_{(1,+\infty)}\bigr),
\end{equation*}
\notag
$$
where $\widehat{\widetilde{h}}(\lambda)=\widehat{h}(-\lambda)$,
$$
\begin{equation*}
h_{-+}=\exp\biggl\{f_-\biggl(\frac{\cdot}{2\pi}\biggr)- f_+\biggl(\frac{\cdot}{2\pi}\biggr)\biggr\},\qquad h_{+-}=\exp\biggl\{f_+\biggl(\frac{\cdot}{2\pi}\biggr)- f_-\biggl(\frac{\cdot}{2\pi}\biggr)\biggr\},
\end{equation*}
\notag
$$
and $\mathfrak{H}(h)$ denotes the Hankel operator acting by the formula
$$
\begin{equation*}
\mathfrak{H}(h)\varphi(s)= \frac{1}{2\pi}\int_0^{+\infty}\widehat{h}(s+t)\varphi(t)\,dt.
\end{equation*}
\notag
$$
Citation in format AMSBIB
\Bibitem{Buf23} |
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