Markov chains of triangles converging to collinearity
The three medians of any triangle $\Delta$ intersect in the
barycenter of $\Delta$ and dissect $\Delta$ into six smaller
children triangles. Let $\Delta_n$ be a Markov chain of
triangles with $\Delta_n$ chosen uniformly among the children of
$\Delta_{n-1}$. P. Diaconis and L. Miclo (2011) show that almost
surely the flatness of $\Delta_n$, namely its maximal edge
length divided by its minimal height length, converges to $\infty$
exponentially fast. In the limit the vertices of $\Delta_n$ become
collinear. D. Mannion (1988, 1990) and S. Volkov (2013) show the same
results for different Markov chains of triangles $\Delta_n$, where the
vertices of $\Delta_n$ are either independently and uniformly chosen in
the interior of $\Delta_{n-1}$ (Mannion) or independently and uniformly
on each edge of $\Delta_{n-1}$ (Volkov). We formulate the above examples
as special cases of the following framework: Suppose that
$A\in\mathbb{C}^{3\times 3}$ is a random matrix with the following two
properties: $(1,1,1)$ is an Eigenvector and for any of the six matrices
$B\in\mathbb{C}^{3\times 3}$ obtained by permuting the columns of the
identity matrix, the random matrices $A$ and $AB$ have the same
distribution. Consider the Markov kernel $M$ on $\mathbb{C}^3$, where
$M(v,\cdot)$ is the distribution of $Av$ in $\mathbb{C}^3$. Let
$V_n\in\mathbb{C}^3$ be a Markov chain with kernel $M$. We identify any
vector $v\in\mathbb{C}^3$ with the triangle whose vertices are the
components of $v$. We prove in particular that almost surely the
flatness of the triangle $V_n$ converges to $\infty$ exponentially fast
with some exact asymptotic rate $\chi > 0$, which only depends on the
distribution of $A$ and not on the starting triangle $V_0$.