By Claire David
Appears in collection : 2026 - T1 - WS2 - Bridging visualization and understanding in Geometry and Topology
We revisit the use of the Weierstrass function as a deterministic driving term for the Schramm--Loewner evolution process (SLE), a setting first considered in earlier work by Joan Lind and Jessica Robbins in 2017. Building on this idea, we introduce a refined deterministic toy model for SLE driven by a Weierstrass drift, based on explicit and sharp estimates for the Lip(delta) norm of the Weierstrass function. Our approach yields improved quantitative bounds -- valid for all delta in ]0,1/2] -- which significantly sharpen the classical Lip(1/2) estimates obtained by truncation methodsby Lind and Robbins. In particular, we determine precise numerical bounds for the critical scaling parameter multiplying the driving term at which the Weierstrass-driven Loewner trace transitions from simple to non-simple curves. We further compare the dynamics generated by prefractal and truncated Weierstrass drifts, showing that the former better captures the roughness, multifractal behaviour, and geometric complexity typically associated with Brownian-driven SLE. These results support the idea that carefully constructed Weierstrass drifts provide a robust and tractable deterministic analogue for studying multifractality and phase transitions in SLE.
By Dalia Terhesiu
By Pierre Arnoux
![[1243] Degrés dynamiques](/media/cache/video_light/uploads/video/SeminaireBourbaki.jpg)