Log-correlated Gaussian Field Research Articles

Critical Gaussian multiplicative chaos for singular measures

Given d≥1, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as e2dXdμ where X is a log-correlated Gaussian field and μ is a locally finite measure on Rd. Our construction generalizes the one performed in the case where μ is the Lebesgue measure. It requires that the measure μ is sufficiently spread out, namely that for μ almost every x we have ∫B(x,1)μ(dy)|x−y|deρlog1|x−y|<∞, where ρ:R+→R+ can be chosen to be any lower envelope function for the 3-Bessel process (this includes ρ(x)=xα with α∈(0,1/2)). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure μ is in a sense optimal.

A remark on Gibbs measures with log-correlated Gaussian fields

Abstract We study Gibbs measures with log-correlated base Gaussian fields on the d-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson’s argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove nonnormalizability of the Gibbs measure. When $d = 2$ , our argument provides an alternative proof of the nonnormalizability result for the focusing $\Phi ^4_2$ -measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein’s inequality on $\mathbb R^d$ . We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) nonnormalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

Multifractal analysis of Gaussian multiplicative chaos and applications

Let $M_{\gamma}$ 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 $M_{\gamma}$ 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.

Almost Surely Time-Space Intermittency for the Parabolic Anderson Model with a Log-Correlated Gaussian Field

Almost Surely Time-Space Intermittency for the Parabolic Anderson Model with a Log-Correlated Gaussian Field

A probabilistic approach of ultraviolet renormalization in the boundary Sine-Gordon model

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field X defined on a subset of \({\mathbb {R}}^d\) by the exponential of its cosine, namely \(\exp (\alpha \smallint \cos (\beta X))\). It has gathered significant attention due to its importance in quantum field theory and to its connection with the study of log-gases in statistical mechanics. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral \(\smallint \cos (\beta X)\) can be defined properly when \(\beta ^2<d\) using the standard Wick normalization of \(\cos (\beta X)\), a more involved renormalization procedure is needed when \(\beta ^2\in [d,2d)\). In particular it exhibits a countable sequence of phase transitions accumulating to the left of \(\beta =\sqrt{2d}\), each transition corresponding to the addition of an extra term in the renormalization scheme. The final threshold \(\beta =\sqrt{2d}\) corresponds to the Kosterlitz–Thouless (KT) phase transition of the \(\log \)-gas. In this paper, we present a novel probabilistic approach to renormalization of the two-dimensional boundary (or 1-dimensional) Sine-Gordon model up to the KT threshold \(\beta =\sqrt{2d}\). The purpose of this approach is to propose a simple and flexible method to treat this problem which, unlike the existing renormalization group techniques, does not rely on translation invariance for the covariance kernel of X or the reference measure along which \(\cos (\beta X)\) is integrated. To this purpose we establish by induction a general formula for the cumulants of a random variable defined on a filtered probability space expressed in terms of brackets of a family of martingales; to the best of our knowledge, the recursion formula is new and might have other applications. We apply this formula to study the cumulants of (approximations of) \(\smallint \cos (\beta X)\). To control all terms produced by the induction procedure, we prove a refinement of classical electrostatic inequalities, which allows us to bound the energy of configurations in terms of the Wasserstein distance between \(+\) and − charges.

The multiplicative chaos of H=0 fractional Brownian fields

We consider a family of fractional Brownian fields {BH}H∈(0,1) on Rd, where H denotes their Hurst parameter. We first define a rich class of normalizing kernels ψ and we rescale the normalised field by the square-root of the gamma function Γ(H), such that the covariance of XH(x)=Γ(H)12(BH(x)−∫RdBH(u)ψ(u,x)du), converges to the covariance of a log-correlated Gaussian field when H↓0. We then use Berestycki’s “good points” approach (Electron. Commun. Probab. 22 (2017) Paper No. 27) in order to derive the convergence of the exponential measure of the fractional Brownian field MγH(dx)=eγXH(x)−γ22E[XH(x)2]dx, towards a Gaussian multiplicative chaos, as H↓0 for all γ∈(0,γ∗(d)), where γ∗(d)> 7 4d. As a corollary we establish the L2 convergence of MγH over the sets of “good points”, where the field XH has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of “good points” with probability close to 1. Moreover, on these sets the volatility converges in L2 to the volatility of multifractal random walks.

Density of imaginary multiplicative chaos via Malliavin calculus

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential mu _beta := :e^{ibeta Gamma (x)}: for a log-correlated Gaussian field Gamma in d ge 1 dimensions. We prove a basic density result, showing that for any nonzero continuous test function f, the complex-valued random variable mu _beta (f) has a smooth density w.r.t. the Lebesgue measure on {mathbb {C}}. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.

A universality result for subcritical complex Gaussian multiplicative chaos

In the present paper, we show that (under some minor technical assumption) Complex Gaussian multiplicative chaos defined as the complex exponential of a log-correlated Gaussian field can be obtained by taking the limit of the exponential of the field convoluted with a smoothing kernel. We consider two types of chaos: eγX for a log correlated field X and γ=α+iβ, α,β∈R and eαX+iβY for X and Y two independent fields with α,β∈R. Our result is valid in the range Psub:={α2+β2<d}∪{|α|∈( d/2, 2d) and |β|< 2d−|α|}, which, up to boundary, is conjectured to be optimal.

The Riemann–Liouville field and its GMC as [formula omitted], and skew flattening for the rough Bergomi model

We consider a re-scaled Riemann–Liouville (RL) process ZtH=∫0t(t−s)H−12dWs, and using Lévy’s continuity theorem for random fields we show that ZH tends weakly to an almost log-correlated Gaussian field Z as H→0. Away from zero, this field differs from a standard Bacry–Muzy field by an a.s.Hölder continuous Gaussian process, and we show that ξγH(dt)=eγZtH−12γ2V ar(ZtH)dt tends to a Gaussian multiplicative chaos (GMC) random measure ξγ for γ∈(0,1) as H→0. We also show convergence in law for ξγH as H→0 for γ∈[0,2) using tightness arguments, and ξγ is non-atomic and locally multifractal away from zero. In the final section, we discuss applications to the Rough Bergomi model; specifically, using Jacod’s stable convergence theorem, we prove the surprising result that (with an appropriate re-scaling) the martingale component Xt of the log stock price tends weakly to Bξγ([0,t]) as H→0, where B is a Brownian motion independent of everything else. This implies that the implied volatility smile for the full rough Bergomi model with ρ≤0 is symmetric in the H→0 limit, and without re-scaling the model tends weakly to the Black–Scholes model as H→0. We also derive a closed-form expression for the conditional third moment E((Xt+h−Xt)3|Ft) (for H>0) given a finite history, and E(XT3) tends to zero (or blows up) exponentially fast as H→0 depending on whether γ is less than or greater than a critical γ≈1.61711 which is the root of 14+12logγ−316γ2. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which Xt/t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t→0.

Reconstructing the base field from imaginary multiplicative chaos

We show that the imaginary multiplicative chaos $\exp(i\beta \Gamma)$ determines the gradient of the underlying field $\Gamma$ 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 $\beta \in (0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t. its imaginary chaos.

Mesoscopic central limit theorem for the circular $\beta $-ensembles and applications

We give a simple proof of a central limit theorem for linear statistics of the circular $\beta $-ensembles which is valid at almost microscopic scales for functions of class $C^{3}$. Using a coupling introduced by Valko and Virag [48], we deduce a central limit theorem for the Sine$_{\beta }$ processes. We also discuss connections between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of [37], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the circular $\beta $-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This establishes that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions coming from log-correlated Gaussian field theory.

The conditional law of the Bacry–Muzy and Riemann–Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces

We compute E(Xt|(Xs)0≤s≤L) for the standard Bacry–Muzy log-correlated Gaussian field X with covariance log+Tt−s, which corrects the finite-horizon prediction formula in Vargas et al. (Duchon et al., 0000). The problem can be viewed as a linear filtering problem, and we solve the problem by showing that the L2(P) closure of {∫[0,L]ϕ(s)Xsds:ϕ∈S,supp(ϕ)⊆[0,L]} is equal to {X(ϕ):ϕ∈H−12,supp(ϕ)⊆[0,L]}, where X(ϕ) is defined as a continuous linear extension of X acting on S⊂Hs, Hs denotes the fractional Sobolev space of order s and P is the law of the field X on the space of tempered distributions. The explicit formula for the filter is obtained as the solution to a Fredholm integral equation of the first kind with logarithmic kernel. From this we characterize the conditional law of the Gaussian multiplicative chaos (GMC) Mγ generated by X, using that Mγ is measurable with respect to X. We also outline how one can adapt this result for the Riemann–Liouville GMC introduced in Forde et al. (2019), which has a natural application to the Rough Bergomi volatility model in the H→0 limit.11We would like to thank Juhan Aru, Janne Junilla, Vincent Vargas and Lauri Vitasaari for helpful discussions.

Universal tail profile of Gaussian multiplicative chaos

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 dimensions.

Decompositions of log-correlated fields with applications

In this article, we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Holder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields whose covariance kernel has a logarithmic singularity on the diagonal—or log-correlated Gaussian fields. The theorem states that any log-correlated Gaussian field $X$ can be decomposed locally into a sum of a Holder continuous function and an independent almost $\star $-scale invariant field (a special class of stationary log-correlated fields with ’cone-like’ white noise representations). This decomposition holds whenever the term $g$ in the covariance kernel $C_{X}(x,y)=\log (1/|x-y|)+g(x,y)$ has locally $H^{d+\varepsilon }$ Sobolev smoothness. We use these decompositions to extend several results that have been known basically only for $\star $-scale invariant fields to general log-correlated fields. These include the existence of critical multiplicative chaos, analytic continuation of the subcritical chaos in the so-called inverse temperature parameter $\beta $, as well as generalised Onsager-type covariance inequalities which play a role in the study of imaginary multiplicative chaos.

Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder

We consider a 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 (Electron. Commun. Probab. 21 (2016) 61) 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 spaces (Ann. Appl. Probab. 26 (2016) 643–690; Adv. Math. 330 (2018) 589–687), and related results for discrete directed polymers (Probab. Theory Related Fields 138 (2007) 391–410; Bates and Chatterjee (2016)), we study the endpoint distribution of a Brownian path under the renormalized GMC measure in this setting. We show that in the low temperature regime, the energy landscape of the system freezes and enters the so-called glassy phase as the entire mass of the Cesaro average of the endpoint GMC distribution stays localized in few spatial islands, forcing the endpoint GMC to be asymptotically purely atomic (Probab. Theory Related Fields 138 (2007) 391–410). The method of our proof is based on the translation-invariant compactification introduced in (Ann. Probab. 44 (2016) 3934–3964) and a fixed point approach related to the cavity method from spin glasses recently used in (Bates and Chatterjee (2016)) in the context of the directed polymer model in the lattice.

Precise high moment asymptotics for parabolic Anderson model with log-correlated Gaussian field

In this paper, we consider the continuous parabolic Anderson model (PAM) driven by a time-independent log-correlated Gaussian field (LGF). We obtain an asymptotic result of Eexp{12∑j,k=1N∫0t∫0tγ(Bj(s)−Bk(r))drds}(N→∞) which is composed of the independent Brownian motions {Bj(s)} and the function γ approximating to a logarithmic potential at 0, such as the covariances of massive free field and Bessel field. Based on the asymptotic result, we get the precise high moment asymptotics for Feynman–Kac formula of the PAM with LGF.

On the Moments of the Moments of the Characteristic Polynomials of Random Unitary Matrices

Denoting by $P_N(A,\theta)=\det(I-Ae^{-i\theta})$ the characteristic polynomial on the unit circle in the complex plane of an $N\times N$ random unitary matrix $A$, we calculate the $k$th moment, defined with respect to an average over $A\in U(N)$, of the random variable corresponding to the $2\beta$th moment of $P_N(A,\theta)$ with respect to the uniform measure $\frac{d\theta}{2\pi}$, for all $k, \beta\in\mathbb{N}$ . These moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log-correlated Gaussian fields. Our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions, and an analysis of a second representation in terms of multiple contour integrals. Our main result is that the moments of moments are polynomials in $N$ of degree $k^2\beta^2-k+1$. This resolves a conjecture of Fyodorov \& Keating~\cite{fyodorov14} concerning the scaling of the moments with $N$ as $N\rightarrow\infty$, for $k,\beta\in\mathbb{N}$. Indeed, it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient.

Liouville metric of star-scale invariant fields: tails and Weyl scaling

We study the Liouville metric associated to an approximation of a log-correlated Gaussian field with short range correlation. We show that below a parameter $$\gamma _c >0$$, the left–right length of rectangles for the Riemannian metric $$e^{\gamma \phi _{0,n}} ds^2$$ with various aspect ratio is concentrated with quasi-lognormal tails, that the renormalized metric is tight when $$\gamma < \min ( \gamma _c, 0.4)$$ and that subsequential limits are consistent with the Weyl scaling.

Convergence of the centered maximum of log-correlated Gaussian fields

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.

Non-universality for first passage percolation on the exponential of log-correlated Gaussian fields

We consider first passage percolation (FPP) where the vertex weight is given by the exponential of two-dimensional log-correlated Gaussian fields. Our work is motivated by understanding the discrete analog for the random metric associated with \emph{Liouville quantum gravity} (LQG), which roughly corresponds to the exponential of a two-dimensional Gaussian free field (GFF). The particular focus of the present paper is an aspect of universality for such FPP among the family of log-correlated Gaussian fields. More precisely, we construct a family of log-correlated Gaussian fields, and show that the FPP distance between two typically sampled vertices (according to the LQG measure) is $N^{1+ O(\epsilon)}$, where $N$ is the side length of the box and $\epsilon$ can be made arbitrarily small if we tune a certain parameter in our construction. That is, the exponents can be arbitrarily close to $1$. Combined with a recent work of the first author and Goswami on an upper bound for this exponent when the underlying field is a GFF, our result implies that such exponent is \emph{not} universal among the family of log-correlated Gaussian fields.