Exponentials of Log-Correlated Gaussian Fields

Exponentials of Log-Correlated Gaussian Fields

Liouville first passage percolation (LFPP) is a way to approximate random distances by exponentiating a regularized log-correlated Gaussian field and using the resulting weights to measure path length. In two dimensions, LFPP is closely tied to the construction and characterization of the LQG metric; for a survey, see (Ding, Dubédat, et al., 2023).

A natural experimental question is what similar exponential metric landscapes look like in higher dimensions. Ding–Gwynne–Zhuang prove tightness for natural exponential-metric approximations in arbitrary dimension in a subcritical regime (Ding, Gwynne, et al., 2023).

This simulation visualizes such metric structures in 3D, including random geodesics and filled metric balls. Use https://aub.ie/lfpp3d for the live demo.

Confluence of random geodesics in 3D

In two-dimensional LQG, unlike in smooth Riemannian geometry, geodesics from two different starting points to the same target can merge before reaching the target and then follow a common final segment. This phenomenon is called confluence, and Gwynne–Miller prove it rigorously for LQG metrics satisfying natural axioms (Gwynne & Miller, 2020).

The following simulation suggests that a similar confluence phenomenon may appear in the 3D exponential-metric model. At this point, it should be read as numerical evidence rather than a continuum theorem. Use https://aub.ie/confluence3d for the confluence demo.

Simulated confluence of random geodesics in a 3D exponential metric landscape.
Ding, J., Dubédat, J., & Gwynne, E. (2023). Introduction to the liouville quantum gravity metric (pp. 4212–4244). https://doi.org/10.4171/icm2022/40
Ding, J., Gwynne, E., & Zhuang, Z. (2023). Tightness of exponential metrics for log-correlated gaussian fields in arbitrary dimension. https://arxiv.org/abs/2310.03996
Gwynne, E., & Miller, J. (2020). Confluence of geodesics in Liouville quantum gravity for \(\gamma \in (0,2)\). Annals Probab., 48(4), 1861–1901. https://doi.org/10.1214/19-AOP1409