Mating of Trees

Use https://aub.ie/mating for the interactive demo. It animates how a pair of random walks can be read as a pair of random trees, and how gluing those trees produces a random surface with fractal geometry.

More precisely, the demo gives a step-by-step view of the mating-of-trees encoding for uniform-spanning-tree-decorated random planar maps. As the time parameter moves, it updates the two-dimensional walk, the left and right frontier lengths, the associated pair of trees, and the explored part of the decorated map in sync.

The continuum analogue replaces the discrete walk by a pair of correlated Brownian motions. This Brownian pair encodes an SLE-decorated Liouville quantum gravity surface, which is the peanosphere, or mating-of-trees, description of LQG (Duplantier et al., 2021; Gwynne et al., 2023).

Duplantier, B., Miller, J., & Sheffield, S. (2021). Liouville quantum gravity as a mating of trees. Astérisque, (427), viii+257. https://doi.org/10.24033/ast

Gwynne, E., Holden, N., & Sun, X. (2023). Mating of trees for random planar maps and Liouville quantum gravity: A survey. In Topics in statistical mechanics (Vol. 59, pp. 41–120). Soc. Math. France, Paris.