Tightness of Liouville first passage percolation for γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$

@article{Ding2019TightnessOL,
  title={Tightness of Liouville first passage percolation for 
$\gamma$
∈
(
0
,
2
)
\$\gamma \in (0,2)\$
},
  author={Jian Ding and Julien Dub'edat and Alexander Dunlap and Hugo Falconet},
  journal={Publications math{\'e}matiques de l'IH{\'E}S},
  year={2019},
  volume={132},
  pages={353-403}
}
We study Liouville first passage percolation metrics associated to a Gaussian free field h $h$ mollified by the two-dimensional heat kernel p t $p_{t}$ in the bulk, and related star-scale invariant metrics. For γ ∈ ( 0 , 2 ) $\gamma \in (0,2)$ and ξ = γ d γ $\xi = \frac{\gamma }{d_{\gamma }}$ , where d γ $d_{\gamma }$ is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020 ), we show that renormalized metrics ( λ t − 1 e ξ p t ∗ h d s ) t… 

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Comparison of discrete and continuum Liouville first passage percolation

  • M. Ang
  • Mathematics
    Electronic Communications in Probability
  • 2019
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
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