How does the supercritical GMC converge?

TL;DR

The paper advances the understanding of supercritical Gaussian multiplicative chaos by providing a stable, process-level description of the convergence of GMC measures built from a star-scale invariant log-correlated field. It achieves this through a detailed local analysis of extremal points, a Brownian-motion based reduction of the shape around mesoscopic maxima, and a resampling framework that yields a canonical limiting shape independent of conditioning thresholds. The main results characterize the limiting objects as a Poissonian superposition of atoms with random weights, governed by a weight process and the (critical) GMC, and they establish a joint Laplace functional that encodes the stable convergence of measure-valued processes across scales. These findings illuminate the freezing phenomenon in the supercritical regime and connect the continuum GMC to a structured, canonical limit akin to a stationary Ornstein–Uhlenbeck-type dynamics, with implications for the uniqueness and robustness of the limiting measure.

Abstract

In the spirit of [M. Biskup & O. Louidor, Adv. Math. 330 (2018)], we study the local structure of $\star$-scale invariant fields -- a class of log-correlated Gaussian fields -- around their extremal points by characterising the law of the "shape" of the field's configuration near such points. As a consequence, we obtain a refined understanding of the freezing phenomenon in supercritical Gaussian multiplicative chaos.

Figures (4)

• Figure 1: The blue curve represents a standard Brownian motion $-B$ run up to time $b$. The light blue region around the Brownian motion is the area enclosed between the curves $[0, b] \ni t \mapsto -B_t - \mathrm{R}_k(t)$ and $[0, b] \ni t \mapsto -B_t + \mathrm{R}_k(t)$ for some $0 \ll k \ll b$. Roughly speaking, Lemma \ref{['lm:approxBrownianBridge']} states that, with high probability, for any $j \in [b-1]$, the supremum of the field $\Upupsilon_{\! b, \mathfrak{g}}(x)$ in the annulus $\mathbb{A}_j$ lies on the vertical segment at $t = j$ within the light blue region.
• Figure 2: A diagram illustrating the decomposition of the cube $[0, 1]^d$ used in the proof of Lemma \ref{['lem:magic']}, in the case $d = 2$. For $\mathrm{R} \geq 1$, we suppose that along each edge of the $d$ edges of $[0, 1]^d$, there are exactly $(e^s+1) (\mathrm{R}+1)^{-1}$ smaller closed boxes, which are drawn in white and that we enumerate by $(A_{i})_{i \in [\mathfrak{I}]}$, for some $\mathfrak{I} \in \mathbb{N}$. For each $i \in [\mathfrak{I}]$, the box $A_{i}$ has side length equal to $R e^{-s}$. The shaded region is the buffer zone $B_{\mathrm{R}, s} = [0, 1]^d \setminus \cup_{i \in [\mathfrak{I}]} A_{i}$. By construction, the buffer zone separates the smaller cubes by a distance of at least $e^{-s}$.
• Figure 3: A typical path in $\mathbb{S}_{t}^{z, A, L}$.
• Figure 4: The point $m$ is chosen inside the small grey ball $\mathbb{B}_{b_t - \log 4}(x)$, and the blue ball represents $\mathbb{B}_{b_t}(m)$. Since we are on the event $\{\mathrm{Y}_t^{\upchi}(m) - \mathfrak{d}_t \geq -A\}$, on the event $\{\mathbb{M}_{m, b_t + 1, b_t} -\mathfrak{d}_t < -A\}$, the supremum of the field $\mathrm{Y}_t^{\upchi}$ inside the large striped ball must coincide with the supremum of the field $\mathrm{Y}_t^{\upchi}$ inside the shaded black ball $\mathbb{B}_{b_t + \log(5/4)}(x)$.

Theorems & Definitions (173)

• Remark 1.1
• Remark 1.2
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Theorem 1.6: Glassy
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Remark 2.1