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 weight $$e^{\xi h(z)}$$ , where h discrete Gaussian free field. We show that distance exponent for LFPP strictly positive all . More precisely, between inner and outer boundaries annulus size $$2^n$$ typically at least $$2^{\alpha n}$$ an $$\alpha depending $$ This crucial input in proof admits non-trivial subsequential scaling limits also has theoretical implications study distances quantum gravity.