Up-to-constants comparison of Liouville first passage percolation and Liouville quantum gravity

Liouville first passage percolation (LFPP) with the parameter ξ > 0 is the family of random distance functions {Dhϵ}ϵ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left\{ {D_h^\epsilon} \right\}_{\, \epsilon > \,0}}$$\end{document} on the plane obtained by integrating eξhϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{e}}^{\xi {h_\epsilon}}}$$\end{document} along paths, where {h∊}∊>0 is a smooth mollification of the planar Gaussian free field. Recent works have shown that for all ξ > 0, the LFPP metrics, appropriately re-scaled, admit non-trivial subsequential limiting metrics. In the case ξ < ξcrit ≈ 0.41, it has been shown that the subsequential limit is unique and defines a metric on γ-Liouville quantum gravity (LQG) γ = γ(ξ) ∈ (0, 2). We prove that for all ξ > 0, each possible subsequential limiting metric is nearly bi-Lipschitz equivalent to the LFPP metric Dhϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_h^\epsilon$$\end{document} when ∊ is small, even if ∊ does not belong to the appropriate subsequence. Using this result, we obtain bounds for the scaling constants for LFPP which are sharp up to polylogarithmic factors. We also prove that any two subsequential limiting metrics are bi-Lipschitz equivalent. Our results are an input in subsequent works which shows that the subsequential limits of LFPP induce the same topology as the Euclidean metric when ξ = ξcrit and that the subsequential limit of LFPP is unique when ξ ≽ ξcrit.