Log-Correlated Gaussian Fields

• Log-correlated Gaussian fields are generalized Gaussian processes defined by logarithmic singularities in their covariance, crucial for modeling critical phenomena.
• They exhibit extreme value statistics with maxima converging to randomly shifted Gumbel distributions and demonstrate Poisson–Dirichlet behavior in low-temperature regimes.
• Their construction via multiplicative chaos and scale decomposition provides a rigorous framework for applications in random geometry, turbulence, finance, and disordered systems.

A log-correlated Gaussian field (LCGF) is a generalized Gaussian process on a domain (typically $\mathbb{R}^d$ or a discrete lattice) in which the covariance of the field behaves logarithmically at small distances. More precisely, if $X$ is such a field, then for two distinct points $x,y$, the covariance function is of the form

$\mathbb{E}[X(x) X(y)] = -\log|x-y| + g(x,y)$

where $g$ is a function that is smooth (or at least bounded) near the diagonal. LCGFs arise in the paper of statistical mechanics, random geometry, random matrices, turbulence, finance, and other areas where long-range spatial correlations and extreme value statistics play a fundamental role.

1. Mathematical Foundations and Definitions

The LCGF is a centered generalized Gaussian process, often only defined as a random distribution due to the logarithmic singularity of its covariance kernel. Its law, up to a global additive constant, is determined by the bilinear form

$$\text{Cov}[(X,\phi_1),(X,\phi_2)] = \iint -\log|y-z|\,\phi_1(y)\phi_2(z)\,dy\,dz$$

for mean-zero test functions $\phi_1, \phi_2$ on $\mathbb{R}^d$ (Duplantier et al., 2014). This structure places LCGFs as the special (“critical”) case $s=d/2$ in the family of fractional Gaussian fields, $X = (-\Delta)^{-s/2} W$, with $W$ being Gaussian white noise and the Hurst index $H = s - d/2 = 0$ for the log-correlated case. In dimensions $d=1$ and $d=2$, the LCGF coincides with the (restriction of the) 2D Gaussian Free Field (GFF), while in higher dimensions it is related to models such as the membrane model and fractional Laplacians.

The logarithmic decay of correlations leads to unique statistical properties and fractal behavior, notably underpinning many “critical” phenomena in random geometry and physics. The field is conformally invariant in every dimension and possesses restriction properties such that, for example, the restriction of a $d$-dimensional LCGF to a lower-dimensional subspace is itself a LCGF on that subspace (Duplantier et al., 2014).

2. Extremes, Maximum, and Extreme Value Theory

The extremal statistics of LCGFs reflect deep structural differences from those of more rapidly decaying (e.g., short-range or power-law) correlated fields.

• Maximum Distribution: The maximum of LCGFs on bounded domains and suitable lattices exhibits tightly controlled probabilistic fluctuations. After recentering by a precisely chosen deterministic sequence $m_n$ or $a_t$, the maximum converges to a randomly shifted Gumbel distribution. In particular, for $\left\{ X_v : v \in V_n \right\}$,

$$\lim_{n \to \infty} \mathbb{P}(M_n - m_n \leq x) = \mathbb{E}\left[\exp(-B^* Z e^{-\sqrt{2d} x})\right],$$

where $Z$ is the limit of a derivative martingale and $B^* > 0$ (Madaule, 2013, Ding et al., 2015, Schweiger et al., 2022). The centering term is typically of the form

$$m_n = \sqrt{2d\log N} - \frac{3}{2\sqrt{2d}} \log\log N + O(1)$$

for a lattice box of side $N$ (Ding et al., 2015).

• Tightness and Tail Estimates: The recentered maximum is tight, and both right and left tails decay exponentially, consistent with explicit estimates

$$\mathbb{P}(\max_{x \in [0,1]^d} X_x^\varepsilon - m_\varepsilon \ge x) \le Ce^{-c x}$$

for suitable mollifications $X_x^\varepsilon$ (Acosta, 2013).

• Universality: The distributional universal nature of the extreme values, and their limits, persists even when the field is defined in random environments or contains rare local defects, provided these are controlled at mesoscopic scales (Schweiger et al., 2022).

3. Gibbs Measure, Low-Temperature Phases, and Poisson-Dirichlet Statistics

Exponentiating the LCGF yields a disordered Boltzmann–Gibbs measure,

$$G_{\beta,N}(x) = \frac{\exp(\beta X_x)}{\sum_{x'\in\mathcal{X}_N} \exp(\beta X_{x'})}$$

with inverse temperature $\beta>0$. The paper of this measure gives rise to phenomena analogous to those found in mean-field spin glasses and random energy models, including glassy and freezing transitions.

• Clustering and Overlaps: In the low-temperature (large $\beta$) regime, the field's high points dominate the measure and the two-overlap distribution collapses to taking values in $\{0,1\}$; specifically,

$$\lim_{N \to \infty} \mathbb{E}[G_{\beta,N}^{\otimes 2}\{q_{12} \le q\}] = \begin{cases} \frac{\beta_c}{\beta} & \text{if } 0 \le q < 1, \ 1 & \text{if } q=1. \end{cases}$$

where $q(x,y)\sim 1$ if $x$ and $y$ are close, and $0$ otherwise (Arguin et al., 2012).

• Poisson–Dirichlet Law: The ordered (normalized) weights of the extreme clusters sampled from the low-temperature Gibbs measure converge in distribution to a Poisson–Dirichlet variable of parameter $\beta_c/\beta$ (with $\beta_c= \sqrt{2}$ or as determined by the field's normalization). This result, proving a conjecture of Carpentier and Le Doussal, demonstrates the Poisson–Dirichlet universality of the glassy phase of LCGFs (Arguin et al., 2012, Ganguly et al., 2023).

4. Gaussian Multiplicative Chaos and Freezing

LCGFs underlie the construction of Gaussian multiplicative chaos (GMC):

$$\mu_\gamma(dx) = \lim_{\epsilon \to 0} \exp(\gamma X_\epsilon(x) - \frac{\gamma^2}{2} \mathbb{E}[X_\epsilon(x)^2])\, dx,$$

where $X_\epsilon$ is a mollification at scale $\epsilon$ and $\gamma$ is a parameter. In the subcritical regime $\gamma^2 < 2d$, this limit yields a nontrivial random measure; at criticality, one must insert an additional normalization such as $\sqrt{\log(1/\epsilon)}$ (Madaule et al., 2013, Duplantier et al., 2014, Vihko, 30 Aug 2024).

• Freezing Transition: When $\gamma^2 > 2d$ or at sufficiently low temperature for the Gibbs measure, the measure $\mu_\gamma$ freezes: its limit is atomic, supported on a discrete set of pure states (clusters of extremal points), and the total measure is dominated by rare high peaks (“glassy phase”) (Madaule et al., 2013, Wong, 2017). The weight of these atoms is governed by the derivative martingale of the field at criticality, and the frozen measure is described as an integrated stable random measure with random intensity given by this derivative martingale.
• Annealed Multifractal Exponents, Pre-freezing: The annaled and quenched scaling exponents for the partition function and multifractal moments exhibit two phase transitions. The “pre-freezing” regime is marked by a transition in the annealed moment exponents before full freezing occurs, with rare localized events dominating higher moments and modifying the thermodynamic behavior of the measure (Wong, 2017).

5. Approximation Schemes, Scale Decomposition, and Field Sample Structure

LCGFs are constructed and analyzed via a variety of representation and approximation schemes:

• Conic and White-Noise Representations: The white-noise construction on a “half-cylinder” (or cone), $\mathcal{C}^+ = [0,1] \times \mathbb{R}_+$, underlies many explicit finite-dimensional models, allowing a decomposition into increments over scales (using, for instance, partitioning by $y$ in the vertical direction) (Arguin et al., 2012, Duplantier et al., 2014).
• Fourier–Sobolev and Chebyshev Expansions: The field can be expanded in bases such as Chebyshev polynomials (in random matrix theory contexts) or fractional Laplace operators in the spectral approach (fractional Gaussian fields) (Duplantier et al., 2014, Webb, 2015).
• Star-scale Invariant Decomposition: Any log-correlated Gaussian field with covariance $C_X(x,y) = \log(1/|x-y|) + g(x,y)$ (with $g$ sufficiently smooth) can be decomposed as $X=L+R$, where $L$ is an almost star-scale invariant field (with explicit cone-like representations or “cascade” structure) and $R$ is independent and almost surely Hölder continuous (Junnila et al., 2018). This decomposition is local (on sufficiently small subdomains).

Construction	Main Structure	Key Use
Half-cylinder/cone	Sum over scales / increments	Extremal process and Gibbs measure analysis
Fourier/Laplace	Spectral/fractional expansion	Generalized functions, functional analysis
*-scale invariant	Cascade, white noise	Multiplicative chaos and critical measures

Such representations are fundamental for the proofs of limit theorems for the maximum, the construction of GMC, and for understanding fine regularity properties of the field and its associated measures.

6. Connections to Applications and Physical Models

LCGFs and their derived measures and metrics are central in a range of modern theoretical and applied topics:

• Random Matrix Theory: LCGFs emerge as scaling limits of global spectral fluctuations in ensembles such as the Gaussian Unitary Ensemble, where the logarithm of the absolute value of the characteristic polynomial and the counting function for eigenvalues are both log-correlated fields (Webb, 2015, Lambert et al., 2016).
• Random Geometry and Liouville Quantum Gravity: In $d=2$, exponentiating the field gives the random geometry of Liouville quantum gravity (LQG), with the Gaussian multiplicative chaos measure providing the area measure, and the exponential metric (involving path integrals with the field) giving the random distance (Duplantier et al., 2014, Hip et al., 2023, Ding et al., 2023). Support theorems and tightness results for exponential metrics illuminate the continuum geometry of random surfaces.
• Statistical Physics and Disordered Systems: The LCGF serves as the energy landscape in versions of random energy models, branching Brownian motion, and spin glass models. The clustering of high points, emergence of Poisson–Dirichlet statistics, and replica symmetry breaking mirror well-known features of mean-field spin glasses (Arguin et al., 2012, Madaule et al., 2013).
• Stochastic PDEs and Fluid Flow in Random Media: The random permeability in pressure equations can be modeled as the Wick exponential of a log-correlated Gaussian field, with existence and uniqueness of solutions established for the renormalized equations, and solutions expressed with GMC measures (Avelin et al., 14 Feb 2024, Avelin et al., 27 May 2024).
• Percolation and High-Dimensional Geometry: In high dimensions, the set of thick points of the LCGF (where the field averages at small scales are large) percolates, forming unbounded connected clusters, in stark contrast to the totally disconnected behavior in $d=2$. This induces new phenomena in associated exponential metrics, such as the possibility of negative set-to-set distance exponents (“super-supercritical” metric phase) (Ding et al., 29 Oct 2024).

7. Extensions, Controversies, and Open Problems

Significant extensions include:

• Reconstruction from Chaos: The values of the field can be reconstructed from its multiplicative chaos measure in arbitrary dimension and even in critical (Seneta–Heyde) regimes, with the scheme covering mildly non-Gaussian fields $X = G + H$ (with $H$ Hölder continuous) (Vihko, 30 Aug 2024).
• Universality and Non-Universality: While extreme value and chaos statistics display remarkable universality across a wide class of LCGFs—even with rare local defects—the geometric scaling exponents of random distances (first passage percolation on exponentials of the field) are non-universal and can be tuned by details of the multiscale decomposition (Ding et al., 2015).
• Metric Tightness and Continuum Limits: The continuum analogues of LQG metrics have been constructed in arbitrary dimensions in the subcritical regime, showing tightness and non-degeneracy, but the uniqueness and precise characterization of the limiting metric law remain open (Ding et al., 2023, Hip et al., 2023).
• Phase Transitions and Percolation: The percolation threshold for thick points undergoes a transition in high dimensions, suggesting distinct geometric phases depending on both thickness parameter and dimension (Ding et al., 29 Oct 2024). The full characterization of critical dimensions, geometric exponents, and universality classes is a subject of ongoing paper.

In summary, log-correlated Gaussian fields constitute a canonical fractal model for spatially extended disordered systems with critical correlations. Their paper has unified and advanced extreme value theory, random geometry, statistical mechanics, and stochastic PDEs, with ongoing developments focusing on the interplay of universality, geometry, and analytical tractability across dimensions and applications.