Probabilistic approach to the analysis of statistics for convex polygonal lines
The author shows that the problem on the statistics of convex polygonal lines solved by \textit{I. Bárány} and \textit{A. M. Vershik} [Geom. Funct. Anal. 2, No. 4, 381--393 (1992; Zbl 0772.52010)] admits an analysis by traditional probabilistic methods. Consider a planar convex polygonal line \(\Gamma\) issuing from zero such that the vertices of \(\Gamma\) are points of the lattice \(Z^2\) and the angle between each side of \(\Gamma\) and the horizontal axis is in the interval \([0, \pi/2]\). The convexity implies that the slopes of the consecutive sides form a strictly increasing sequence. The space of such polygonal lines is denoted by \(\Pi\). Let \(X\) be the set of all pairs of coprime positive integers. Lemma. The space \(\Pi\) is in one-to-one correspondence with the space \(C_0 (X)\) of finite nonnegative integer-valued functions \(v(x)\) on \(X\) \((v(x)\) is said to be finite if \(v(x) \neq 0\) only for finitely many \(x)\). This lemma implies that \(\Gamma \in \Pi\) may be thought of as finite configurations of ideal gas of particles at point of \(X\) [see also \textit{A. M. Vershik} [Funct. Anal. Appl. 28, No. 1, 13--20 (1994); translation from Funkts. Anal. Prilozh. 28, No. 1, 16--25 (1994; Zbl 0848.52004)].
scientific article; zbMATH DE number 795615