KPZ relation does not hold for the level lines and the SLE${}_{\kappa }$ flow lines of the Gaussian free field

Abstract

In this paper we mingle the Gaussian free field, the Schramm-Loewner evolution and the KPZ relation in a natural way, shedding new light on all of them. Our principal result shows that the level lines and the SLE${}_{\kappa }$ flow lines of the Gaussian free field do not satisfy the usual KPZ relation. In order to prove this, we have to make a technical detour: by a careful study of a certain diffusion process, we provide exact estimates of the exponential moments of winding of chordal SLE curves conditioned to pass nearby a fixed point. This extends previous results on winding of SLE curves by Schramm.

arXiv comment47 pages; 4 beautiful images and 2 other figures; in ver2: extended SLE winding theorem 5.1 to also cover the case κ= 4; minor revisions all over the paper