Numerical estimation of the Hausdorff dimension of D-random feuilletages

TL;DR

This paper develops numerical methods to estimate the Hausdorff dimension of D-random feuilletages, providing results consistent with known and conjectured dimensions for specific cases, advancing understanding of higher-dimensional random geometries.

Contribution

It introduces a numerical approach using finite-size scaling to estimate the Hausdorff dimension of D-random feuilletages, a new class of higher-dimensional random geometries.

Findings

Estimated Hausdorff dimension for D=2 matches known results for the Brownian map.

Numerical estimates for D=3 align with the conjectured dimension of 8.

Method demonstrates effectiveness for higher-dimensional random geometries.

Abstract

We implement numerical techniques to simulate D-random feuilletages, candidates for higher-dimensional random geometries introduced in L. Lionni and J.-F. Marckert, Math. Phys. Anal. Geom. 24 (2021) 39. Using finite-size scaling techniques, our approach allows to give a numerical estimation of the Hausdorff dimension $d_H$ of these feuilletages. The results obtained are compatible with the formal result known for the Brownian map, which corresponds to the D=2 random feuilletage. For the D=3 case, our numerical study finds a good agreement with the conjectured value $d_H=8$.