Raoul Robert, Vincent Vargas
In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow.
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Inverse of the Gaussian multiplicative chaos: Moments
本文系统研究了高斯乘性混沌逆过程的一般性质与单点矩，为该领域提供了理论基础。
In this article, some properties of complex Wiener-Itô multiple integrals and complex Ornstein-Uhlenbeck operators and semigroups are obtained. Those include Stroock's formula, Hu-Meyer formula, Clark-Ocone formula and the hypercontractivity of complex Ornstein-Uhlenbeck semigroups. As an application, several expansions of the fourth moments of complex Wiener-Itô multiple integrals are given.
This review-style article presents an overview of recent progress in constructing and studying critical Gaussian multiplicative chaos. A proof that the critical measure in any dimension can be obtained as a limit of subcritical measures is given.
We propose a new definition of the Gaussian multiplicative chaos (GMC) and an approach based on the relation of subcritical GMC to randomized shifts of a Gaussian measure. Using this relation we prove general uniqueness and convergence results for subcritical GMC that hold for Gaussian fields with arbitrary covariance kernels.
We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos.
We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series \[\sum_{k=1}^\infty \frac{1}{\sqrt{k}}(A_k \cos(2πk x) + B_k \sin(2πk x)),\] where $A_…
In this article we study the tail probability of the mass of Gaussian multiplicative chaos. With the novel use of a Tauberian argument and Goldie's implicit renewal theorem, we provide a unified approach to general log-correlated Gaussian fields in arbitrary dimension and derive precise first order asymptotics of the tail probability, resolving a conjecture of Rhodes and Vargas. The leading order is described by a universal constant that captures the generic property of Gaussian multiplicative chaos, and may be seen as the analogue of the Liouville unit volume reflection coefficients in higher…
Irreducibility of polynomials with random multiplicative coefficients revisited
本文针对取值于{±1}的随机完全乘法函数构成的多项式，证明其不可约概率的上极限随次数趋于无穷时等于1，推进了随机乘法系数多项式不可约性的研究。
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(γ< \sqrt{2d})$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field)
Subcritical multiplicative chaos and the characteristic polynomial of the C$β$E
本文利用circular beta-ensembles的可积性，证明了其特征多项式与特征值计数函数在整个次临界阶段（含负幂次）的乘性混沌收敛，将近期酉情形的结果推广至任意beta>0。
Inverse of the Gaussian multiplicative chaos: an integration by parts formula
本文在高斯乘性混沌及其逆的框架下研究分部积分公式的对应版本，推广了高斯过程击中时中的相关结果。
The standard augmented multiplicative coalescent revisited
本文针对Erdős-Rényi随机图连续时间版本的标准增广乘法coalescent标度极限，采用同时广度优先游走的图探索过程，提出了更自洽简单直接的构建方法，且该方法便于推广到非标准情形。
We show that for $γ<\sqrt{4/3}$, it is possible to define the Levy area of a planar Brownian motion with the Liouville measure of intermittency parameter $γ$ as the underlying area measure. We also consider the case of smoother curves, and study some properties of the integration map thus defined.
We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.
The Fourier coefficients of the holomorphic multiplicative chaos in the limit of large frequency
本文研究全纯乘法混沌的大频率 Fourier 系数，证明归一化后依分布收敛于由 Gaussian 乘法混沌总质量缩放的复正态变量，并推广到连续系数的过程收敛，推出 circular-β-系综次线性特征系数的收敛律。
Absolute continuity of non-Gaussian and Gaussian multiplicative chaos measures
本文研究了非高斯和高斯乘积混沌测度的绝对连续性，通过构造耦合证明两者在几乎处处相互绝对连续。
The real layering field of Brownian loop soup and the Gaussian multiplicative chaos
本文研究复平面有界单连通域中布朗环汤的分层场，证明其经重整化后收敛到亚临界高斯乘性混沌，主要技术为维纳-伊藤混沌展开，并计算了$n$点函数及其共形协变性和边界行为。
In this article we study the tail probability of the mass of critical Gaussian multiplicative chaos (GMC) associated to a general class of log-correlated Gaussian fields in any dimension, including the Gaussian free field (GFF) in dimension two. More precisely, we derive a fully explicit formula for the leading order asymptotics for the tail probability and demonstrate a new universality phenomenon. Our analysis here shares similar philosophy with the subcritical case but requires a different approach due to complications in the analogous localisation step, and we also employ techniques from r…
Convergence of processes time-changed by Gaussian multiplicative chaos
本文在$L^2$区域中证明了由高斯乘性混沌进行时间变换的过程的收敛性，并给出Liouville简单随机游走和Liouville $α$-稳定过程的收敛实例。
We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.
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 regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.
The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.
The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos
本文证明临界二维随机热流不属于高斯乘性混沌类，通过推导其矩的严格下界实现，该下界具有独立意义。
Brownian multiplicative chaos measures, introduced in [Jeg20, AHS20, BBK94], are random Borel measures that can be formally defined by exponentiating $γ$ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $γ$ is less than 2 were studied. This article considers the critical case where $γ=2$, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation…
Let $M_γ$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_γ$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.
The random trigonometric series $\sum_{n=1}^\infty ρ_n \cos (nt +ω_n)$ on the circle $\mathbb{T}$ are studied under the conditions $\sum |ρ_n|^2=\infty$ and $ρ_n\to 0$, where $\{ω_n\}$ are iid and uniformly distributed on $\mathbb{T}$. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which are the limits of the exponentiations of partials sums. which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on…
The density of imaginary multiplicative chaos is positive
本文证明了虚乘积混沌的密度在复平面上处处严格正，为高斯空间上其他函数的研究提供了新方法。
In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purp…
The weak solution of Tanaka's SDE is not a function of the driving Brownian motion, and therefore it has no Wiener chaos expansion. However in some sense explained here it has a generalised chaos expansion involving infinite products of stochastic differentials accumulating at the minimum of the Brownian path. This is related to the existence of a non-classical noise richer than the usual white noise.
Denote by $μ_β="\exp(βX)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset\mathbb{R}^d$. The case $β\in\mathbb{R}$ has been studied quite intensively, and then $μ_β$ is a random measure on $U$. It is known that $μ_β$ can also be defined for complex values $β$ lying in certain subdomain of $\mathbb{C}$, and then the realizations of $μ_β$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $β$ was considered) by studying the Besov-regularity of $μ_β…
We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereb…
Decoupling and Multipoint moments for the Inverse of the Gaussian multiplicative chaos
本文研究了高斯乘性混沌逆的解耦结构与多点矩，作为扩展随机共形焊接工作至逆Lehto焊接存在性的初步研究，并证明了正实轴上逆同胚的伸缩率属于$L^{1}([0,1]\ imes[0,2])$。
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.
Hölder type estimates for Gaussian multiplicative chaos
本文研究了GMC比率的右尾行为，通过扩展GMC比率类推导出其上界和下界，利用启发式方法证明最优指数。
A theory of intermittency differentiation is developed for a general class of 1D Infinitely Divisible Multiplicative Chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman-Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in i…
We prove convergence of a stochastic approximation of powers of the Riemann $ζ$ function to a non-Gaussian multiplicative chaos measure, and prove that this measure is a non-trivial multifractal random measure. The results cover both the subcritical and critical chaos. A basic ingredient of the proof is a 'good' Gaussian approximation of the induced random fields that is potentially of independent interest.
Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.
On small deviations of Gaussian multiplicative chaos with a strictly logarithmic covariance on Euclidean ball
本文证明了在严格对数协方差核的正定性区域内，GMC的小偏差始终为对数正态型，并探讨了其在Laplace变换语境下的显式常数界限与新表示。
Multiplicative chaos from random multiplicative functions
本研究从扭曲斯坦豪斯乘性函数的狄利克雷级数构造乘性混沌测度，通过新颖部分求和技巧证明其普适性，还证明其与GMC测度几乎必然相互绝对连续，统一扩展了Gorodetsky与Wong的相关结果。
We consider a {\it{Gaussian multiplicative chaos}} (GMC) measure on the classical Wiener space driven by a smoothened (Gaussian) space-time white noise. For $d\geq 3$ it was shown in \cite{MSZ16} that for small noise intensity, the total mass of the GMC converges to a strictly positive random variable, while larger disorder strength (i.e., low temperature) forces the total mass to lose uniform integrability, eventually producing a vanishing limit. Inspired by strong localization phenomena for log-correlated Gaussian fields and Gaussian multiplicative chaos in the finite dimensional Euclidean s…
Uniqueness of supercritical Gaussian multiplicative chaos
本文研究了超临界高斯乘法混沌测度的唯一性，证明其收敛到非平凡极限，且该极限仅由正则化选择决定。
Random analytic functions via Gaussian multiplicative chaos
本研究以高斯乘性测度为Clark测度定义单位圆盘上的随机解析函数$\varphi$，证明其几乎必然为Blaschke乘积，并给出零点密度的精确估计。
Sharp inequalities involving multiplicative chaos sums
本文通过引理和方法获得通用极值不等式，并证明由乘法系统生成的函数系统收敛。
A theory of intermittency differentiation is developed for a general class of Gaussian Multiplicative Chaos measures including the measure of Bacry and Muzy on the interval and circle as special cases. An exact, non-local functional equation is derived for the derivative of a general functional of the total mass of the measure with respect to intermittency. The formal solution is given in the form of an intermittency expansion and proved to be a renormalized expansion in the centered moments of the total mass of the measure. The full intermittency expansion of the Mellin transform of the total…
In this note we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole $L^1$- or subcritical phase of the chaos measure.
Bulk/boundary quotients of Gaussian multiplicative chaos measures II: Tail profile of bulk Gaussian multiplicative chaos measures in the exact scale-invariant case
本文研究精确尺度不变核下边界具有均匀奇异性的高斯乘性混沌测度的右尾轮廓，通过将局部化技巧应用于边界上的辅助测度，推广了 Rhodes-Vargas 和 Wong 的平坦背景几何结果，并用体/边界商的变体刻画尾部轮廓系数。
Characterisation of planar Brownian multiplicative chaos
本文通过特征化平面布朗乘法混沌测度，揭示了随机游走厚点集的标度极限，并得出了归一化后厚点数量的收敛性结果。
This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter $γ^2$ beyond the transition phase (i.e. $γ^2>2d$) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and …
We discuss a Gaussian multiplicative chaos (GMC) structure underlying a family of random measures $\mathbf{M}_r$, indexed by $r\in\mathbb{R}$, on a space $Γ$ of directed pathways crossing a diamond fractal with Hausdorff dimension two. The laws of these random continuum path measures arise in a critical weak-disorder limiting regime for discrete directed polymers on disordered hierarchical graphs. For the analogous subcritical continuum polymer model in which the diamond fractal has Hausdorff dimension less than two, the random path measures can be constructed as subcritical GMCs through coupl…
The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self-contained, and perhaps somewhat more accessible.
We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.
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 as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.
We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computatio…
Almost sure upper bound for a model problem for multiplicative chaos in number theory
本研究在简化的函数域类似设定下证明了数论乘法混沌模型问题的新几乎必然上界，其结果与作者此前关于随机乘法函数几乎必然上界的结论相当，且因研究对象更简单，证明更易理解。
Inverse of the Gaussian multiplicative chaos: Lehto welding of Independent Quantum disks
本文利用随机共形焊接框架，证明了在γ<0.1818条件下高斯乘性混沌的逆及其独立副本的Lehto焊接存在性，并得到粘合两个GMC边界圆盘的共形不变环Γ，但尚未证明其为SLE环。
We show that the imaginary multiplicative chaos $\exp(iβΓ)$ determines the gradient of the underlying field $Γ$ for all log-correlated Gaussian fields with covariance of the form $-\log |x-y| + g(x,y)$ with mild regularity conditions on $g$, for all $d \geq 2$ and for all $β\in (0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t. its imaginary chaos.
Almost sure upper bound for sums of random multiplicative functions and critical chaos
该研究运用临界混沌理论方法，改进了Steinhaus与Rademacher型随机乘性函数部分和的几乎必然上界，以强形式证明了Harper的相关猜想并确定了精确对数指数。
We prove that if $ω$ is uniformly distributed on $[0,1]$, then as $T\to\infty$, $t\mapsto ζ(iωT+it+1/2)$ converges to a non-trivial random generalized function, which in turn is identified as a product of a very well behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between number theory and the theory of multiplicative chaos -- the latter is known to be connected to many other areas of mathematics. We also investigate the statistical behavior of the zeta function on the …
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 closely linked to ideas in probability theory connected with multiplicative chaos; another such closely related problem is the Fyodorov-Hiary-Keating conjecture on the maximum size of the Riemann zeta function in intervals of bounded length on the critical line. In this paper we consider a problem that might be thought of as a simplified function field version…
The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.
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