Let \(\lambda\) be a probability measure on the standard unit disk of the complex plane \({\mathbb{C}}\). The authors randomly assign a complex number in the unit disk for each cell in a square grid in the complex plane according to the measure \(\lambda\). The collection of these numbers defines a Beltrami coefficient \(\mu(z)\) on \({\mathbb{C}}\) which is constant on the cells of the grid. The Beltrami equation \(\bar\partial w(z)=\mu(z)\partial w(z)\) has a unique injective solution \(w^\mu\) that fixes \(0\), \(1\) and \(\infty\). The authors call \(w^\mu\) a random quasiconformal mapping, but point out that \(w^\mu\) may not be quasiconformal. The first main result of the paper says that if the mesh size of the grid is small, then with high probability, \(w^{\mu}\) is close to an affine transformation \(A_\lambda\) determined by the measure \(\lambda\).
A circle packing is a collection of circles in \({\mathbb{C}}\) with disjoint interiors. By the Koebe-Andreev-Thurston circle packing theorem, any finite triangulation of a topological disk admits a maximal circle packing whose boundary circles are horocycles. A discrete set \(V\) of points in \({\mathbb{C}}\) determines a Voronoi tessellation. This means that \({\mathbb{C}}\) can be written as a union of sets \(F_x\), \(x\in V\), where \(F_x\) consists of all points \(z\in {\mathbb{C}}\) for which \(\min_{y\in V}\vert y-z\vert=\vert x-z\vert\). If the points in \(V\) are in general position, then the Delauney triangulation is the dual graph to the Voronoi tessellation. The union of all the triangles in the Delauney triangulation is the convex hull of \(V\), and hence a topological disk.
Let \(\Omega\subset {\mathbb{C}}\) be a simply connected domain bounded by a \(\mathrm{C}^1\)-curve. The authors randomly choose \(N\geq 1\) points in \(\Omega\) with respect to a Lebesgue measure. For technical reasons, they also choose \(\asymp \sqrt{N}\) equally spaced points on the boundary \(\partial\Omega\).They define the random Delauney triangulation as the union of the Delauney triangles contained in \(\Omega\). Let \(\varphi_{\mathcal{P}}\) denote the circle packing map of the suitably normalized maximal circle packing of the random Delauney triangulation. Kenneth Stephenson has suggested that when \(N\) is large, then with high probability, \(\varphi_{\mathcal{P}}\) approximates a conformal map \(\varphi\colon \Omega \to {\mathbb{D}}\). The second main result of the paper is a proof of this conjecture.