Related papers: Random Curves by Conformal Welding

In this paper we give a general family of conformal invariants associated to bordered Riemann surfaces endowed with boundary parametrizations, or equivalently compact surfaces endowed with conformal maps. Each invariant is specified by a…

This manuscript explores the connections between a class of stochastic processes called "Stochastic Loewner Evolution" (SLE) and conformal field theory (CFT). First some important results are recalled which we utilise in the sequel, in…

The conformal loop ensemble CLE$_\kappa$ with parameter $8/3 < \kappa < 8$ is the canonical conformally invariant measure on countably infinite collections of non-crossing loops in a simply connected domain. We show that the number of loops…

A surface with boundary is randomly generated by gluing polygons along some of their sides. We show that its genus and number of boundary components asymptotically follow a bivariate normal distribution.

We consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve…

We show how to efficiently count and generate uniformly at random finitely generated subgroups of the modular group $\textsf{PSL}(2,\mathbb{Z})$ of a given isomorphism type. The method to achieve these results relies on a natural map of…

(a) We propose a ``static'' construction procedure for random networks with given correlations of the degrees of the nearest-neighbor vertices. This is an equilibrium graph, maximally random under the constraint that its degree-degree…

We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We…

We study the relationship between certain SLE$_\kappa(\rho)$ processes, which are variants of the Schramm-Loewner evolution with parameter $\kappa$ in which one keeps track of an extra marked point, and Liouville quantum gravity (LQG).…

We derive recursive equations for the characteristic numbers of rational nodal plane curves with at most one cusp, subject to point conditions, tangent conditions and flag conditions, developing techniques akin to quantum cohomology on a…

We obtain simple characterizations of the connected components of the space of closed curves on the 2-sphere whose geodesic curvatures are constrained to lie in an open interval $(\kappa_1,\kappa_2)$, in terms of $\kappa_1$ and $\kappa_2$.…

We suggest an invariant way to enumerate nodal and nodal-cuspidal real deformations of real plane curve singularities. The key idea is to assign Welschinger signs to the counted deformations. Our invariants can be viewed as a local version…

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high…

In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which are used in computer aided geometric design. We consider these curves in the framework of the similarity geometry and characterize…

In low dimensions, conformal anomaly has profound influence on the critical behavior of random surfaces with extrinsic curvature rigidity $1/\a$. We illustrate this by making a small $D$ expansion of rigid random surfaces, where a…

We show that when one draws a simple conformal loop ensemble (CLE$_\kappa$ for $\kappa \in (8/3,4)$) on an independent $\sqrt{\kappa}$-Liouville quantum gravity (LQG) surface and explores the CLE in a natural Markovian way, the quantum…

We construct and study the conformal loop ensembles CLE(kappa), defined for all kappa between 8/3 and 8, using branching variants of SLE(kappa) called exploration trees. The conformal loop ensembles are random collections of countably many…

This article gives a comprehensive description of the fractal geometry of conformally-invariant (CI) scaling curves, in the plane or half-plane. It focuses on deriving critical exponents associated with interacting random paths, by…

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties.…

We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as…