Percolation of thick points of the log-correlated Gaussian field in high dimensions

TL;DR

The paper addresses the percolative geometry of thick points for the log-correlated Gaussian field in high dimensions, proving that for fixed $\alpha>0$ the thick-point set $\mathscr{T}_\alpha$ contains an unbounded path when $d$ is sufficiently large. The main method combines discrete path constructions on coarse lattices with first- and second-moment arguments to produce $\alpha$-thick paths, then passes to subsequential limits to obtain continuous thick paths; the authors extend this strategy to BRW and white-noise fields and show thick-paths persist under different approximations. A key implication is for the exponential metric, where in high dimensions and large $\xi$ the set-to-set distance exponent $\widetilde{Q}(\xi)$, if it exists, can be negative, signaling a new phase distinct from the 2D Liouville quantum gravity regime. The work also connects to fractal percolation by showing that the critical probability $p_c(d)$ tends to zero as $d\to\infty$, and it establishes a unified thick-point geometry across several regularizations, highlighting a dimension-driven transition in the geometry of extreme values of log-correlated fields.

Abstract

We prove that the set of thick points of the log-correlated Gaussian field contains an unbounded path in sufficiently high dimensions. This contrasts with the two-dimensional case, where Aru, Papon, and Powell (2023) showed that the set of thick points is totally disconnected. This result has an interesting implication for the exponential metric of the log-correlated Gaussian field: in sufficiently high dimensions, when the parameter $ξ$ is large, the set-to-set distance exponent (if it exists) is negative. This suggests that a new phase may emerge for the exponential metric, which does not appear in two dimensions. In addition, we establish similar results for the set of thick points of the branching random walk. As an intermediate result, we also prove that the critical probability for fractal percolation converges to 0 as $d \to \infty$.

Figures (5)

• Figure 1: A self-avoiding path of length 11 in a tube $T_{x,y}$ with $x = 0$ and $y = 8 \mathbf e_1$ (the two green points). Its starting point has $|\cdot|_1$-distance 1 to $x$, and its ending point has $|\cdot|_1$-distance 1 to $y$.
• Figure 2: The colored paths are the refinements of $(x_i, x_{i+1})$ for $1 \leq i \leq 6$. In this figure, the refinements of neighboring edges intersect only at their endpoints and do not intersect at other vertices (though they may appear to intersect at other vertices due to the two-dimensional illustration).
• Figure 3: The colored paths $P_{l-1}$ and $Q_{l-1}$ are defined on $8^{-l+1} \mathbb{Z}^d$, and the colored paths $P_l$ and $Q_l$ are defined on $8^{-l} \mathbb{Z}^d$. The shaded region corresponds to a $2^{-j}$-box with $j = 3l-2$ that intersects both $P_l$ and $Q_l$. We can map it to a pair $(x,e)$ such that $e$ is an edge in $P_{l-1}$ that intersects $Q_{l-1}$, and $x$ is a vertex in $P_l$ that belongs to the refinement of $e$. The red point corresponds to $x$, and the dashed red edge corresponds to $e$.
• Figure 4: The green path $P$ is defined on $\frac{1}{\mathfrak r} 8^{-j} \mathbb{Z}^d$, and the blue path $P_{j-s}$ is defined on $\frac{1}{\mathfrak r} 8^{-j+s} \mathbb{Z}^d$. If we choose the integer $s$ such that $r < \frac{3}{14} 8^s$, then all the vertices in $P \cap \overline{B_{\frac{r}{\mathfrak r} 8^{-j}}(x)}$ belong to the refinements of at most two edges in $P_{j-s}$ at scale $s$, as shown by the dashed red edges in the figure. This gives an upper bound on $|P \cap \overline{B_{\frac{r}{\mathfrak r} 8^{-j}}(x)}|$.
• Figure 5: There exists a half-space $\mathbb{H}$ such that ${\rm vol}(B_t(y) \cap B_t(u) \cap (x_u + [-\frac{1}{2}, \frac{1}{2}]^d)) \geq {\rm vol}(B_t(u) \cap B_t(y) \cap \mathbb{H})$. The shaded domain corresponds to $B_t(u) \cap B_t(y) \cap \mathbb{H}$, whose volume is lower-bounded by some positive universal constant times ${\rm vol}(B_t(0))$.

Theorems & Definitions (72)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• proof
• Theorem 1.6
• Theorem 1.7
• Remark 1.8: Super-supercritical exponential metric
• Proposition 2.1