Asymptotic behavior of Fermat distances in the presence of noise.

Resumen:  Fermat distances are metrics designed for datasets supported on a manifold. These distances are given by geodesics in the weighted graph determined by the points in which long jumps are penalized. When the points are given by a Poisson Point Process in Euclidean spaces, this model coincides with Euclidean First Passage Percolation (Howard-Newman 1997). In both contexts it is natural to consider perturbations of the model. We consider such perturbations and prove that if the noise converges to zero, then the noisy microscopic Fermat distance converges to the non-noisy macroscopic Fermat distance. In the Euclidean case, this corresponds to a continuity result for the time constant.

Posted on Sep 23, 2024 in Seminario de Probabilidades de Chile, Seminars