[Paper Review] Tightness of supercritical Liouville first passage percolation

Jian Ding, Ewain Gwynne|arXiv (Cornell University)|May 27, 2020

Stochastic processes and statistical mechanics43 references4 citations

TL;DR

This paper establishes the tightness of re-scaled Liouville first passage percolation (LFPP) metrics in the supercritical phase ($\xi > \xi_{\mathrm{crit}}$), proving that all subsequential limits are complete, non-Euclidean metrics where most points are isolated with infinite pairwise distances. The results suggest a connection to Liouville quantum gravity with matter central charge in $(1,25)$, extending prior work on the subcritical phase.

ABSTRACT

Liouville first passage percolation (LFPP) with parameter $ξ>0$ is the family of random distance functions $\{D_h^ε\}_{ε>0}$ on the plane obtained by integrating $e^{ξh_ε}$ along paths, where $h_ε$ for $ε>0$ 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 $ξ_{\mathrm{crit}} > 0$ such that for $ξ< ξ_{\mathrm{crit}}$, 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 $γ$-\emph{Liouville quantum gravity metric} for $γ= γ(ξ)\in (0,2)$). We show that for all $ξ> 0$, the LFPP metrics are tight with respect to the topology on lower semicontinuous functions. For $ξ> ξ_{\mathrm{crit}}$, every possible subsequential limit $D_h$ is a metric on the plane which does \emph{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 $(1,25)$.

Motivation & Objective

• To establish tightness of re-scaled Liouville first passage percolation (LFPP) metrics in the supercritical phase ($\xi > \xi_{\mathrm{crit}}$).
• To characterize the structure of subsequential limits of LFPP metrics when $\xi > \xi_{\mathrm{crit}}$, particularly their topological and geometric properties.
• To extend the known convergence results from the subcritical phase ($\xi < \xi_{\mathrm{crit}}$) to the supercritical regime, where the limiting metric does not induce the Euclidean topology.
• To connect the limiting behavior of LFPP in the supercritical phase to Liouville quantum gravity with matter central charge in $(1,25)$.

Proposed method

• Use of a subadditivity argument to establish the existence of the distance exponent $Q(\xi)$ for all $\xi > 0$, which governs the scaling of the normalizing constant $\mathfrak{a}_\varepsilon$.
• Application of Efron-Stein-type concentration inequalities to control fluctuations in left-right crossing distances across dyadic annuli in the plane.
• Establishment of moment estimates and bounds on distances around and across annuli to control the variance of LFPP distances at different scales.
• Comparison of $D_h^\varepsilon$ with a white-noise approximation of the GFF to transfer estimates from a simpler model to the full LFPP setting.
• Use of Gaussian concentration and tail estimates (e.g., Lemma A.2) to bound the exponential moments of the field $h_\varepsilon^*$, crucial for controlling the metric's behavior.
• Proof of lower semicontinuity and triangle inequality for the limiting metric $D_h$, ensuring it is a well-defined extended metric on $\mathbb{C}$.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of Liouville first passage percolation (LFPP) in the supercritical phase ($\xi > \xi_{\mathrm{crit}}$)?
• RQ2 Do the re-scaled LFPP metrics $\mathfrak{a}_\varepsilon^{-1} D_h^\varepsilon$ admit subsequential scaling limits in the supercritical regime?
• RQ3 What topological and geometric properties do the subsequential limits of LFPP exhibit when $\xi > \xi_{\mathrm{crit}}$?
• RQ4 How does the structure of the limiting metric differ from the subcritical case, particularly in terms of completeness and the topology it induces?
• RQ5 Is there a connection between the supercritical LFPP limits and Liouville quantum gravity with matter central charge in $(1,25)$?

Key findings

• For all $\xi > 0$, the re-scaled LFPP metrics $\mathfrak{a}_\varepsilon^{-1} D_h^\varepsilon$ are tight with respect to the topology of lower semicontinuous functions on $\mathbb{C} \times \mathbb{C}$.
• In the supercritical phase ($\xi > \xi_{\mathrm{crit}}$), every subsequential limit $D_h$ is a complete extended metric that does not induce the Euclidean topology.
• For $\xi > \xi_{\mathrm{crit}}$, there exists an uncountable, dense, Lebesgue measure-zero set of points $z \in \mathbb{C}$ such that $D_h(z,w) = \infty$ for all $w \neq z$, indicating that most points are isolated.
• The distance exponent $Q(\xi)$ exists and satisfies $\mathfrak{a}_\varepsilon = \varepsilon^{1 - \xi Q(\xi) + o_\varepsilon(1)}$ as $\varepsilon \to 0$, with $Q(\xi)$ continuous, strictly decreasing on $(0, 0.7)$, and non-increasing on $(0, \infty)$.
• The critical value $\xi_{\mathrm{crit}}$ satisfies $0.4135 \leq \xi_{\mathrm{crit}} \leq 0.4189$, and lies in the supercritical phase for $\xi > \xi_{\mathrm{crit}}$, where the limiting metric is non-Euclidean.
• The authors expect the supercritical limits to correspond to Liouville quantum gravity with matter central charge in $(1,25)$, extending the known correspondence in the subcritical phase.