相关论文: On the regularity of complex multiplicative chaos
The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary, and more generally…
We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well…
We construct and study properties of an infinite dimensional analog of Kahane's theory of Gaussian multiplicative chaos \cite{K85}. Namely, if $H_T(\omega)$ is a random field defined w.r.t. space-time white noise $\dot B$ and integrated…
Multifractal systems usually have singularity spectra defined on bounded sets of H\"older exponents. As a consequence, their associated multifractal scaling exponents are expected to depend linearly upon statistical moment orders at high…
We find the precise rate at which the empirical measure associated to a $\beta$-ensemble converges to its limiting measure. In our setting the $\beta$-ensemble is a random point process on a compact complex manifolds distributed according…
We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from the Circular $\beta$-Ensemble, which are defined as the coefficients of $\{z^n\}$ in the characteristic polynomial $\det(1-zU_{N}^{*})$. When…
Recognizing the regime of positive definiteness for a strictly logarithmic covariance kernel, we prove that the small deviations of a related Gaussian multiplicative chaos (GMC) $M_\gamma$ are for each natural dimension $d$ always of…
Consider a logarithmically-correlated Gaussian field $X$ in $d$ dimensions. For all $\gamma \in (-\sqrt{2d},\sqrt{2d})$, we show that the derivatives $\frac{\partial^k}{\partial\gamma^k} :e^{\gamma X_\epsilon}:$ of the regularised Gaussian…
We introduce and carefully study a natural probability measure over the numerical range of a complex matrix $A \in M_n(\C)$. This numerical measure $\mu_A$ can be defined as the law of the random variable $<AX,X> \in \C$ when the vector $X…
The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical…
For an $N \times T$ random matrix $X(\beta)$ with weakly dependent uniformly sub-Gaussian entries $x_{it}(\beta)$ that may depend on a possibly infinite-dimensional parameter $\beta\in \mathbf{B}$, we obtain a uniform bound on its operator…
The holomorphic multiplicative chaos (HMC) is a holomorphic analogue of the Gaussian multiplicative chaos. It arises naturally as the limit in large matrix size of the characteristic polynomial of Haar unitary matrices, and more generally,…
We propose a new diagnostic for quantum chaos. We show that time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order…
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane's seminal work in 1985. Though this beautiful paper faded from memory until recently, it already contains ideas and results that are…
Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity area measure $\mu := e^{\gamma h(z)}…
We consider $N\times N$ matrices $X$ with independent, identically distributed entries, and prove that the sequence of measures $\frac{ | \det (X-z)|^\gamma}{\mathbb{E}[ | \det (X-z)|^\gamma]}$ converge to the Gaussian Multiplicative Chaos…
A new characterization of random fields appearing in physical models is presented that is based on their well-known Homogeneous Chaos expansions. We take advantage of the adaptation capabilities of these expansions where the core idea is to…
Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is…
A new class of particle systems with sequential interaction is proposed to approximate the McKean-Vlasov process that originally arises as the limit of the mean-field interacting particle system. The weighted empirical measure of this…
In this article, we consider the multiplicative chaos measure associated to the log-correlated random Fourier series, or random wave model, with i.i.d. coefficients taken from a general class of distributions. This measure was shown to be…