Intersections of SLE Paths: the double and cut point dimension of SLE

TL;DR

This paper calculates the Hausdorff dimensions of double points and cut points of SLE_kappa for kappa > 4, confirming longstanding predictions and extending results to various SLE processes and flow line intersections.

Contribution

It provides the first rigorous computation of the Hausdorff dimensions of double and cut points of SLE_kappa for kappa > 4, including new results for radial and whole-plane SLE and flow line intersections.

Findings

Hausdorff dimension of double points of SLE_kappa for kappa > 4

Hausdorff dimension of cut points of SLE_kappa for kappa > 4

Dimensions of intersections of flow lines of Gaussian free field

Abstract

We compute the almost-sure Hausdorff dimension of the double points of chordal SLE_kappa for kappa > 4, confirming a prediction of Duplantier-Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal SLE_kappa for kappa > 4 as well as analogous dimensions for the radial and whole-plane SLE_kappa(rho) processes for kappa > 0. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field e^{ih/chi}, where h is a Gaussian free field and chi > 0, of different angles with each other and with the domain boundary.