Bounds on the distance exponent for higher-dimensional Liouville first passage percolation

TL;DR

This paper extends the study of LFPP to higher dimensions $(d\ge 3)$ by defining the random metric $D_h^{\varepsilon,\xi}$ via a mollified log-correlated Gaussian field and analyzing the set-to-set distance exponent $\lambda(d,\xi)$. It establishes universal bounds for $\lambda(d,\xi)$ and differential constraints on $\lambda'(d,\xi)$, proves that $\lambda$ is nondecreasing with dimension, and develops a subcritical fractal-dimension theory through the unique $\mathsf d_\gamma$ solving $\lambda(d,\xi)=1-\xi Q$ with $Q=d/\gamma+\gamma/2$, showing that $\mathsf d_\gamma$ is continuous and strictly increasing in $\gamma$ with explicit bounds. Collectively, these results constitute the first quantitative distance-exponent bounds in higher dimensions and lay groundwork toward a higher-dimensional LQG-type geometry, while supporting prior assumptions in related work. The work also provides a framework for comparing mollifications and connects the dimension-growth behavior to fractal properties of LFPP in the subcritical regime.

Abstract

For $ξ\geq 0$ and $d \geq 3$, the higher-dimensional Liouville first passage percolation (LFPP) is a random metric on $ε\mathbb{Z}^d$ obtained by reweighting each vertex by $e^{ξh_ε(x)}$, where $h_ε(x)$ is a continuous mollification of the whole-space log-correlated Gaussian field. This metric generalizes the two-dimensional LFPP, which is related to Liouville quantum gravity. We derive several estimates for the set-to-set distance exponent of this metric, including upper and lower bounds and bounds on its derivative with respect to $ξ$. In the subcritical region for $ξ$, we derive estimates for the fractal dimension and show that it is continuous and strictly increasing with respect to $ξ$. In particular, our result is an important step towards proving a technical assumption made in previous work by the first author and Gwynne. These are also the first bounds on the distance exponent for LFPP in higher dimensions.

Figures (2)

• Figure 1: Left: Graph of the upper and lower bounds on $\lambda(2,\xi)$ from Theorem \ref{['thm:bound-2d']}. By the connection between $\sqrt{8/3}$-LQG and the Brownian map, we have $\lambda(2, 1/\sqrt{6}) = 1/6$. Right: Graph of the upper and lower bounds on $\lambda(3,\xi)$ from Theorem \ref{['thm:bound']}. We do not know the explicit value of $\lambda(3, \xi)$ for any $\xi>0$.
• Figure 2: Graph of the upper and lower bounds on $\mathsf d_\gamma$ from Corollary \ref{['cor']}. Left: The case $d = 3$. Right: The case $d = 4$.

Theorems & Definitions (21)

• Remark 1.1
• Theorem 1.2: gp-lfppang-discrete-lfpp
• Theorem 1.3
• Lemma 1.4
• proof : Proof of Theorem \ref{['thm:bound']} given Lemma \ref{['lem:dimen-restriction']}
• Theorem 1.5
• proof
• Lemma 1.6
• Definition 1.7
• Proposition 1.8