The distance exponent for Liouville first passage percolation is positive

Discrete Liouville first passage percolation (LFPP) with parameter $\xi > 0$ is the random metric on a sub-graph of $\mathbb Z^2$ obtained by assigning each vertex $z$ a weight of $e^{\xi h(z)}$, where $h$ is the discrete Gaussian free field. We show that the distance exponent for discrete LFPP is strictly positive for all $\xi > 0$. More precisely, the discrete LFPP distance between the inner and outer boundaries of a discrete annulus of size $2^n$ is typically at least $2^{\alpha n}$ for an exponent $\alpha > 0$ depending on $\xi$. This is a crucial input in the proof that LFPP admits non-trivial subsequential scaling limits for all $\xi > 0$ and also has theoretical implications for the study of distances in Liouville quantum gravity.

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