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

We study Liouville first passage percolation metrics associated to a Gaussian free field $h$ mollified by the two-dimensional heat kernel $p_t$ in the bulk, and related star-scale invariant metrics. For $γ∈ (0 , 2)$ and $ξ=γ/d_γ$, where $d_γ$ is the Liouville quantum gravity dimension defined in Ding and Gwynne (Commun. Math. Phys. 374:1877–1934, 2020), we show that renormalized metrics $(λ_t^{−1}e^{ξp_t∗h}ds)_{t∈(0,1)}$ are tight with respect to the uniform topology. We also show that subsequential limits are bi-Hölder with respect to the Euclidean metric, obtain tail estimates for side-to-side distances, and derive error bounds for the normalizing constants $λ_t$.

Duke Scholars

Author Alexander J Dunlap Mathematics