The critical Liouville quantum gravity metric induces the Euclidean topology

TL;DR

This paper proves that the critical Liouville quantum gravity metric induces the same topology as the Euclidean metric, with a specific modulus of continuity, confirming the metric's topological compatibility with the plane.

Contribution

It establishes that all subsequential limits of critical Liouville first passage percolation induce the Euclidean topology, with a precise modulus of continuity.

Findings

Critical LQG metric induces Euclidean topology

Optimal modulus of continuity is a power of 1/ log(1/|x|)

Results apply to subsequential limits of LQG approximations

Abstract

We show that every possible metric associated with critical ($\gamma=2$) Liouville quantum gravity (LQG) induces the same topology on the plane as the Euclidean metric. More precisely, we show that the optimal modulus of continuity of the critical LQG metric with respect to the Euclidean metric is a power of $1/\log(1/|\cdot|)$. Our result applies to every possible subsequential limit of critical Liouville first passage percolation, a natural approximation scheme for the LQG metric which was recently shown to be tight.