Work

Perimeter length of the convex hull of Brownian motion in the hyperbolic plane

Year: 2025

Type: preprint

Abstract: We relate the expected hyperbolic length of the perimeter of the convex hull of the trajectory of Brownian motion in the hyperbolic plane to an expectation of a certain exponential functional of a one-dimensional real-valued Brownian motion, and hence derive small- and large-time asymptotics for the expected hyperbolic perimeter. In contrast to the case of Euclidean Brownian motion with non-zero drift, the large-time asymptotics are a factor of two greater than the lower bound implied by the fact that the convex hull includes the hyperbolic line segment from the origin to the endpoint of the hyperbolic Brownian motion. We also obtain an exact expression for the expected perimeter length after an independent exponential random time.

Source: arXiv (Cornell University)

Authors: Bhattacharjee, Chinmoy, Versendaal, Rik, Wade, Andrew

Institutions: Universität Hamburg, Delft University of Technology, Durham University

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Deutsche Forschungsgemeinschaft, Engineering and Physical Sciences Research Council

Awards: Isaac Newton Institute for Mathematical Sciences (INI), Superkonzentration in zufälligen geometrischen Strukturen, Anomalous diffusion via self-interaction and reflection, G6527098369

OpenAlex ID: W6979327433

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