Loop-Erased Random Surfaces

Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2 dimensional spanning tree and declaring that LERS should have the same relation to these 2 trees as LERW has to ordinary spanning trees. Furthermore we present numerical evidence that the growth rate for LERS on a $\delta$ fine grid as $\delta \to 0$ is $2.5269 \pm 0.0017$ and we hypothesize that it has an exact value of 48/19. This suggests the possibility of a fractal limiting object for LERS analogous to SLE for LERW.