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Exponentially fast convergence to flat triangles in the iterated barycentric subdivision Klöckner, Matthias

Abstract

The barycentric subdivision dissects a triangle along its three medians into six children triangles. The children of a flat triangle (i.e. the vertices are collinear) are flat. For any triangle Δ let the shape S(Δ) be the unique complex point z in the first quadrant such that Δ is similar to the triangle −1, 1, z in which the edge between −1 and 1 has maximal length. Only flat triangles’ shapes lie on the real line. If Δ^(n) is a Markov chain of triangles with Δ^(n) chosen uniformly amongst the children of Δ^(n−1), then we call the Markov chain S(Δ^(n)) a shape chain and we call it (non-)flat, if Δ(⁰) and therefore each Δ^(n) is (non-)flat. Let Zn be a non-flat and Xn be a flat shape chain. We say that a sequence of random variables Wn taking values in ℂ \ {0} decays exactly or at least with rate X, if X > 0 and almost surely limn1/n*ln|Wn| = −X or lim supn1/nln |Wn| ≤ −X, resp. In a paper from 2011, P. Diaconis and L. Miclo show that =Zn decays at least with rate X' for some universal constant X' and that Xn has an invariant measure μ. We prove that =Zn decays exactly with rate X for a universal constant X which we express as an integral w.r.t. μ. The above paper also shows the convergence of Zn − Xn to 0 in probability for a specific coupling (Xn,Zn). For this coupling we prove that Zn − Xn decays exactly with rate X.

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