Triangular-grid billiards and plabic graphs

Abstract

Given a polygon \(P\) in the triangular grid, we obtain a permutation \(\pi_P\) via a natural billiards system in which beams of light bounce around inside of \(P\). The different cycles in \(\pi_P\) correspond to the different trajectories of light beams. We prove that \[\operatorname{area}(P)\geq 6\operatorname{cyc}(P)-6\quad\text{and}\quad\operatorname{perim}(P)\geq\frac{7}{2}\operatorname{cyc}(P)-\frac{3}{2},\] where \(\operatorname{area}(P)\) and \(\operatorname{perim}(P)\) are the (appropriately normalized) area and perimeter of \(P\), respectively, and \(\operatorname{cyc}(P)\) is the number of cycles in \(\pi_P\). The inequality concerning \(\operatorname{area}(P)\) is tight, and we characterize the polygons \(P\) satisfying \(\operatorname{area}(P)=6\operatorname{cyc}(P)-6\). These results can be reformulated in the language of Postnikov's plabic graphs as follows. Let \(G\) be a connected reduced plabic graph with essential dimension \(2\). Suppose \(G\) has \(n\) marked boundary points and \(v\) (internal) vertices, and let \(c\) be the number of cycles in the trip permutation of \(G\). Then we have \[v\geq 6c-6\quad\text{and}\quad n\geq\frac{7}{2}c-\frac{3}{2}.\]Mathematics Subject Classifications: 05D99, 51M04Keywords: Triangular grid, billiards, plabic graph, membrane