Abstract
Discrete and continuum Liouville first passage percolation (DLFPP, LFPP) are two approximations of the conjectural $\gamma$-Liouville quantum gravity (LQG) metric, obtained by exponentiating the discrete Gaussian free field (GFF) and the circle average regularization of the continuum GFF respectively. We show that these two models can be coupled so that with high probability distances in these models agree up to $o(1)$ errors in the exponent, and thus have the same distance exponent. Ding and Gwynne (2018) give a formula for the continuum LFPP distance exponent in terms of the $\gamma$-LQG dimension exponent $d_\gamma$. Using results of Ding and Li (2018) on the level set percolation of the discrete GFF, we bound the DLFPP distance exponent and hence obtain a new lower bound $d_\gamma \geq 2 + \frac{\gamma^2}2$. This improves on previous lower bounds for $d_\gamma$ for the regime $\gamma \in (\gamma_0, 0.576)$, for some small nonexplicit $\gamma_0 > 0$.
Highlights
Let h be a continuum Gaussian free field (GFF) on a connected domain D ⊂ C.For γ ∈ (0, 2], the γ-Liouville quantum gravity (γ-LQG) surface is, heuristically speaking, the random two-dimensional Riemannian manifold with metric given by eγh(dx2 + dy[2]).This definition does not make literal sense as h is a distribution, but by using regularization procedures one can make sense of the random volume form of γ-LQG [Kah[85], DS11, RV14]
For γ ∈ (0, 2], the γ-Liouville quantum gravity (γ-LQG) surface is, heuristically speaking, the random two-dimensional Riemannian manifold with metric given by eγh(dx2 + dy2)
An important problem is to understand the metric structure of γ-LQG
Summary
Let h be a continuum Gaussian free field (GFF) on a connected domain D ⊂ C. For γ ∈ (0, 2], the γ-Liouville quantum gravity (γ-LQG) surface is, heuristically speaking, the random two-dimensional Riemannian manifold with metric given by eγh(dx2 + dy[2]). Dunlap-Falconet [DDDF19], who show that for γ ∈ (0, 2) one can take a regularization of γ-LQG called continuum Liouville first passage percolation (LFPP), and obtain a metric space by sending the regularization parameter to zero along a subsequential limit[1]. A discrete analog of continuum LFPP is discrete Liouville first passage percolation (DLFPP), in which one s√amples a discrete Gaussian free field (DGFF) η on an n × n lattice, assigns a weight of eξ π 2 η(v) to each vertex v, and defines the distance between two vertices to be the weight of the minimum-weight path between the vertices. We do not use any results from that series of works, so there is no cyclic dependence
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
