Dubejko, Tomasz Recurrent random walks, Liouville’s theorem and circle packings. (English) Zbl 0888.30005 Math. Proc. Camb. Philos. Soc. 121, No. 3, 531-546 (1997). Let \(\mathbb{K}\) be a simplicial 2-complex given by a triangulation of a domain in the complex plane \(\mathbb{C}\). A collection \(\mathcal P\) of circles in \(\mathbb{C}\) is called a circle packing for \(\mathbb{K}\) if there exists a 1-to-1 correspondence \(v\leftrightarrow C(v)\) between the vertices of \(\mathbb{K}\) and the centres of the circles of \(\mathcal P\) such that the triangles of \(\mathbb{K}\) correspond to mutually touching circles of \(\mathcal P\) in an orientation preserving way. If all circles of \(\mathcal P\) have mutually disjoint interiors, \(\mathcal P\) is called univalent.Let \(\mathcal P\) and \(\mathcal Q\) be circle packings for \(\mathbb{K}\). The cp-map from \(\mathcal P\) to \(\mathcal Q\) is essentially a function which maps each center of a circle in \(\mathcal P\) to the center of the corresponding circle in \(\mathcal Q\) and extends via barycentric coordinates. The ratio function from \(\mathcal P\) to \(\mathcal Q\) gives for each vertex of \(\mathbb{K}\) the ratio of the radii of the corresponding circles in \(\mathcal P\) and \(\mathcal Q\) and is again extended via barycentric coordinates.The paper relates circle packings to the concept of recurrent graphs, which is defined in the following way. The simple random walk on a graph is given by associating to each pair of vertices \((v,w)\) the transition probability \(1/\delta(v)\) if \(w\) is a neighbour of \(v\) and \(0\) otherwise, where \(\delta(v)\) denotes the degree of \(v\). The graph is called recurrent if the simple random walk visits every vertex infinitely many times with probability 1.One of the main results is the following discrete analogue of Liouville’s theorem for ratio functions. Let \(\mathcal P\) be a univalent circle packing such that the corresponding triangulation covers the whole plane \(\mathbb{C}\). Suppose \(f\) is a cp-map from \(\mathcal P\) to \(\mathcal Q\), and \(f^\#\) is the associated ratio function. If the 1-skeleton of the complex of \(\mathcal P\) is recurrent then either \(f^\#\) is unbounded or is constant.The paper investigates in particular branched circle packings, where there are vertices \(v\) in \(\mathbb{K}\) such that the chain of circles associated with the neighbors of \(v\) winds several times around \(C(v)\). The set of all of these vertices is called the branch set. The following result concerns the uniqueness of branched circle packings. Let \(\mathbb{K}\) be a bounded degree triangulation of an open disc with recurrent 1-skeleton, and \(F_{\mathfrak B}\) be the set of all bounded valence circle packings for \(\mathbb{K}\) with finite branch set \(\mathfrak B\). If \(F_{\mathfrak B}\neq\emptyset\) then all packings in \(F_{\mathfrak B}\) are copies of each other under similarities of \(\mathbb{C}\). Reviewer: J.Linhart (Salzburg) Cited in 1 ReviewCited in 4 Documents MSC: 30C20 Conformal mappings of special domains 30C62 Quasiconformal mappings in the complex plane 60G50 Sums of independent random variables; random walks 52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry) Keywords:recurrent graphs; branched circle packings; circle packing map; ratio map PDFBibTeX XMLCite \textit{T. Dubejko}, Math. Proc. Camb. Philos. Soc. 121, No. 3, 531--546 (1997; Zbl 0888.30005) Full Text: DOI arXiv