Tightness of Liouville first passage percolation for ${\hat{I}}^3\in (0,2)$

Abstract

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 ${\hat{I}}^3\in (0,2)$ and $\hat{I}¾=\frac{\hat{I}}{³}{d}_{\hat{I}}^3$, where $d_{\hat{I}}^3$ is the Liouville quantum gravity dimension defined in [Ding-Gwynne18], we show that renormalized metrics $(\hat{I}{»}_t^{-1}{e}^{\hat{I}¾p_t\ast h}ds{)}_{t\in (0,1)}$ are tight with respect to the uniform topology. In particular, we show that subsequential limits are bi-Hölder with respect to the Euclidean topology, obtain tail estimates for side-to-side distances and derive error bounds for the normalizing constants $\hat{I}{»}_t$.

39 pages, 3 figures. To appear in Publ. Math. IHÉS