01. September 2025

Planar maps with high degrees

Figure A random spanning-tree decorated triangulation of the torus with 100000 triangles viewed via its periodic Tutte embedding in the plane.

Figure Boltzmann planar map with parameter a = 2.45

Figure Boltzmann planar map with parameter a = 2.35

Figure Boltzmann planar map with parameter a = 2.3

Figure Boltzmann planar map with parameter a = 2.0

Figure Boltzmann planar map with parameter a = 1.8

Figure Boltzmann planar map with parameter a = 1.7

References

[1] Budd, Timothy, and Nicolas Curien. “Geometry of infinite planar maps with high degrees.” (2017): 1-37.

[2] Bertoin, Jean, Timothy Budd, Nicolas Curien, and Igor Kortchemski. “Martingales in self-similar growth-fragmentations and their connections with random planar maps.” Probability Theory and Related Fields 172, no. 3 (2018): 663-724.

[3] Budd, Timothy, Nicolas Curien, and Cyril Marzouk. “Infinite random planar maps related to Cauchy processes.” Journal de l’École polytechnique-Mathématiques 5 (2018): 749-791.

[4] Budd, Timothy. “Lessons from the mathematics of two-dimensional Euclidean quantum gravity.” In Handbook of Quantum Gravity, pp. 1-55. Singapore: Springer Nature Singapore, 2023.