The planar SHE describes heat flow or random polymers on an inhomogeneous surface. It is a finite-temperature version of planar first passage percolation such as the Eden growth model.

It is the first model with plane symmetries for which we can show convergence to the directed landscape.

The methods use a Skorokhod integral representation and Gaussian multiplicative chaos on path space. Joint work with Jeremy Quastel and Alejandro Ramirez.