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

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