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**Minkowski content and natural parameterization for the Schramm-Loewner evolution.**
*(English)*
Zbl 1331.60165

In this important and impressive paper, the authors continue their study of natural parametrization for Schramm-Loewner evolution (\(\mathrm{SLE}\)). Earlier work in this direction was carried out by Lawler, Sheffield, Zhou, and others. The authors establish existence of the \(d\)-dimensional Minkowski content for \(\mathrm{SLE}_{\kappa}\) (for \(d=1+\kappa/8\) and \(\kappa< 8\)), and are then able to use the Minkowski content to define a natural parametrization. While existence of a natural parametrization was established in earlier works, the approach in this paper has the advantage that several important properties of natural parametrization become easier to derive.

The half-plane capacity parametrization of \(\mathrm{SLE}\) curves arises naturally in the context of the Loewner equation, but that parametrization is less natural from the point of view of discrete models whose scaling limits can be described using \(\mathrm{SLE}\); for the discrete curves arising from such processes, there is often an intrinsic notion of “length” that one might hope would persist in the scaling limit as well.

The paper features a well-written introduction that explains the motivation for the search for a natural parametrization, and gives an overview of earlier work. Several objects that are important in their own right, such as Green’s functions for \(\mathrm{SLE}\), are discussed. Following the introduction, a brief outline of the paper is provided, and the basic ideas that go into the proofs are presented. The proofs themselves feature a powerful blend of analytic and probabilistic techniques.

The half-plane capacity parametrization of \(\mathrm{SLE}\) curves arises naturally in the context of the Loewner equation, but that parametrization is less natural from the point of view of discrete models whose scaling limits can be described using \(\mathrm{SLE}\); for the discrete curves arising from such processes, there is often an intrinsic notion of “length” that one might hope would persist in the scaling limit as well.

The paper features a well-written introduction that explains the motivation for the search for a natural parametrization, and gives an overview of earlier work. Several objects that are important in their own right, such as Green’s functions for \(\mathrm{SLE}\), are discussed. Following the introduction, a brief outline of the paper is provided, and the basic ideas that go into the proofs are presented. The proofs themselves feature a powerful blend of analytic and probabilistic techniques.

Reviewer: Alan A. Sola (Cambridge)

### MSC:

60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B27 | Critical phenomena in equilibrium statistical mechanics |

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\textit{G. F. Lawler} and \textit{M. A. Rezaei}, Ann. Probab. 43, No. 3, 1082--1120 (2015; Zbl 1331.60165)

### References:

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