In this talk I will give a brief introduction to Liouville first-passage percolation (LFPP) which is a model for random metric on a finite planar grid graph. It was studied primarily as a way to understand the random metric associated with Liouville quantum gravity (LQG), one of the major open problems in contemporary probability theory. In short the Liouville quantum gravity is a (conjectured) one parameter family of ``canonicalâ€™â€™ random metrics on a Riemann surface. I will discuss some recent results on this metric and the main focus will be on estimates of the typical distance between two points. I will highlight the apparent disagreement of these estimates with a prediction made in the physics literature about the LQG metric. I will also mention some (of many) future problems in this program. Based on a joint work with Jian Ding.

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