On the smoothness of the partition function for multiple Schramm-Loewner evolutions

TL;DR

This paper proves that the partition function for multiple $SLE_$ curves is twice continuously differentiable when <, using derivative estimates to establish smoothness.

Contribution

The authors provide a direct proof of the $C^2$ smoothness of the partition function for multiple $SLE_$ curves when <, advancing understanding of their regularity properties.

Findings

Partition function is $C^2$ for <.

Derivative estimates are key to establishing smoothness.

The proof offers a new direct approach to regularity of $SLE$ partition functions.

Abstract

We consider the measure on multiple chordal Schramm-Loewner evolution ($SLE_\kappa$) curves. We establish a derivative estimate and use it to give a direct proof that the partition function is $C^2$ if $\kappa<4$.