Liouville First Passage Percolation Research Articles

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 \({\left\{ {D_h^\epsilon} \right\}_{\, \epsilon > \,0}}\) on the plane obtained by integrating \({{\rm{e}}^{\xi {h_\epsilon}}}\) 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 \(D_h^\epsilon\) 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.

Tightness of supercritical Liouville first passage percolation

Liouville first passage percolation (LFPP) with parameter $\xi >0$ is the family of random distance functions ${D\_h^\epsilon}{\epsilon >0}$ on the plane obtained by integrating $e^{\xi h\epsilon}$ along paths, where $h\_\epsilon$ for $\epsilon >0$ is a smooth mollification of the planar Gaussian free field. Previous work by Ding–Dubédat–Dunlap–Falconet and Gwynne–Miller has shown that there is a critical value $\xi\_{\mathrm{crit}} > 0$ such that for $\xi < \xi\_{\mathrm{crit}}$, LFPP converges under appropriate re-scaling to a random metric on the plane which induces the same topology as the Euclidean metric (the so-called $\gamma$-Liouville quantum gravity metric for $\gamma = \gamma(\xi)\in (0,2)$). We show that for all $\xi > 0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $\xi > \xi\_{\mathrm{crit}}$, every possible subsequential limit $D\_h$ is a metric on the plane which does not induce the Euclidean topology: rather, there is an uncountable, dense, Lebesgue measure-zero set of points $z\in\mathbb C$ such that $D\_h(z,w) = \infty$ for every $w\in\mathbb C\setminus {z}$. We expect that these subsequential limiting metrics are related to Liouville quantum gravity with matter central charge in $(1,25)$.

The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter \(\xi > 0\) is the random metric on a sub-graph of \(\mathbb Z^2\) obtained by assigning each vertex z a weight of \(e^{\xi h(z)}\), where h is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all \(\xi > 0\). More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size \(2^n\) is typically at least \(2^{\alpha n}\) for an exponent \(\alpha > 0\) depending on \(\xi \). This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all \(\xi > 0\) and also has theoretical implications for the study of distances in Liouville quantum gravity.

Existence and uniqueness of the Liouville quantum gravity metric for gamma in (0,2)

We show that for each gamma in (0,2), there is a unique metric (i.e., distance function) associated with gamma -Liouville quantum gravity (LQG). More precisely, we show that for the whole-plane Gaussian free field (GFF) h, there is a unique random metric D_h associated with the Riemannian metric tensor “e^{gamma h} (dx^2 + dy^2)” on {mathbb {C}} which is characterized by a certain list of axioms: it is locally determined by h and it transforms appropriately when either adding a continuous function to h or applying a conformal automorphism of mathbb {C} (i.e., a complex affine transformation). Metrics associated with other variants of the GFF can be constructed using local absolute continuity. The gamma -LQG metric can be constructed explicitly as the scaling limit of Liouville first passage percolation (LFPP), the random metric obtained by exponentiating a mollified version of the GFF. Earlier work by Ding et al. (Tightness of Liouville first passage percolation for gamma in (0,2), 2019. arXiv:1904.08021) showed that LFPP admits non-trivial subsequential limits. This paper shows that the subsequential limit is unique and satisfies our list of axioms. In the case when gamma = sqrt{8/3}, our metric coincides with the sqrt{8/3}-LQG metric constructed in previous work by Miller and Sheffield, which in turn is equivalent to the Brownian map for a certain variant of the GFF. For general gamma in (0,2), we conjecture that our metric is the Gromov–Hausdorff limit of appropriate weighted random planar map models, equipped with their graph distance. We include a substantial list of open problems.

Liouville first-passage percolation: Subsequential scaling limits at high temperature

Let $\{Y_{\mathfrak{B}}(x):x\in\mathfrak{B}\}$ be a discrete Gaussian free field in a two-dimensional box $\mathfrak{B}$ of side length $S$ with Dirichlet boundary conditions. We study Liouville first-passage percolation: the shortest-path metric in which each vertex $x$ is given a weight of $e^{\gamma Y_{\mathfrak{B}}(x)}$ for some $\gamma>0$. We show that for sufficiently small but fixed $\gamma>0$, for any sequence of scales $\{S_{k}\}$ there exists a subsequence along which the appropriately scaled and interpolated Liouville FPP metric converges in the Gromov–Hausdorff sense to a random metric on the unit square in $\mathbf{R}^{2}$. In addition, all possible (conjecturally unique) scaling limits are homeomorphic by bi-Hölder-continuous homeomorphisms to the unit square with the Euclidean metric.

Bounds for distances and geodesic dimension in Liouville first passage percolation

For $\xi \geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb{Z} ^{2}$ obtained by weighting each vertex by $\varepsilon e^{\xi h_{\varepsilon }(z)}$, where $h_{\varepsilon }(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_{\varepsilon }(z)$. Ding and Gwynne (2018) showed that for $\gamma \in (0,2)$, LFPP with parameter $\xi = \gamma /d_{\gamma }$ is related to $\gamma $-Liouville quantum gravity (LQG), where $d_{\gamma }$ is the $\gamma $-LQG dimension exponent. For $\xi > 2/d_{2}$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $\xi \geq 0$. For $\xi \leq 2/d_{2}$, this leads to new bounds for $d_{\gamma }$ which improve on the best previously known upper (resp. lower) bounds for $d_{\gamma }$ in the case when $\gamma > \sqrt{8/3} $ (resp. $\gamma \in (0.4981, \sqrt{8/3} )$). These bounds are consistent with the Watabiki (1993) prediction for $d_{\gamma }$. However, for $\xi > 1/\sqrt{3} $ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki’s prediction to the $\xi >2/d_{2}$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.