On the plant leaf's boundary, 'jupe àgodets' and conformal embeddings

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 àgodets' (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 a = 1/2 from the exponentially growing to the flat structure.