2001/07/31 by Sergei Nechaev, Raphaël Voituriez, Raphael Voituriez · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Boundary (topology) #Conformal map #Curvature #Embedding #Euclidean geometry #Euclidean space #Exponent #Geometry #Mathematical analysis #Mathematics #Point processes and geometric inequalities #Pure mathematics #Stochastic processes and statistical mechanics #Surface (topology) #Topological and Geometric Data Analysis #Tree (set theory) #cond-mat.soft #cond-mat.stat-mech #math.MG
paper · pdf · doi:10.1088/0305-4470/34/49/322
17 pages (revtex), 8 eps-figures, to appear in Journal of Physics A
arxiv created 2001/11/19 · openalex publication_date 2001/12/05 · openalex created_date 2016/06/24 · arxiv updated 2016/08/31 · openalex updated_date 2026/08/05
The stable profile of the boundary of a plant's leaf fluctuating in the direction transverse to the leaf's surface is described in the framework of a model called a `surface à godets' (SG). It is shown that the information on the profile is encoded in the Jacobian of a conformal mapping (the coefficient of deformation) corresponding to an isometric embedding of a uniform Cayley tree into the 3D Euclidean space. The geometric characteristics of the leaf's boundary (such as the perimeter and the height) are calculated. In addition, a symbolic language allowing us to investigate the statistical properties of a SG with annealed random defects of the curvature of density q is developed. It is found that, at q = 1, the surface exhibits a phase transition with the critical exponent α = ½ from the exponentially growing to the flat structure.