01. October 2025

Random surface: bipolar oriented triangulation

Figure A bipolar-oriented triangulation with 20000 triangles.

Figure A bipolar-oriented triangulation with 30000 triangles.

Figure A bipolar-oriented triangulation with 40000 triangles.

Figure Another large random bipolar-oriented triangulation.

Video A bipolar-oriented triangulation with 20000 triangles.

Video A large random bipolar oriented triangulation of the 2-sphere.

References

[1] Kenyon, Richard; Miller, Jason; Sheffield, Scott; Wilson, David B. “Bipolar orientations on planar maps and SLE12”. Ann. Probab. 47 (2019), no. 3, 1240–1269.

[2] Barkley, Jerome, and Timothy Budd. “Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity.” Classical and Quantum Gravity 36, no. 24 (2019): 244001.

[3] Gwynne, Ewain, Nina Holden, and Xin Sun. “Joint scaling limit of a bipolar-oriented triangulation and its dual in the peanosphere sense.” arXiv preprint arXiv:1603.01194 (2016).