A Phase Transition For Repeated K-Averages

Let $x_1,\dots,x_{n}$ be a fixed sequence of real numbers. At each stage, pick $k$ integers $\{I_{i}\}_{1\leq i \leq k}$ uniformly at random without replacement and then for each $i \in \{1,2,\dots,k\}$ replace $x_{I_i}$ by $(x_{I_1}+x_{I_2}+\dots+x_{I_k})/k$. It is easy to observe that all the co-ordinates converge to $(x_1+\dots+x_n)/n$. In this article, we extend the result of \cite{chatterjee2019note} by establishing order of decay of the expected $L^{2}$ distance. Furthermore, we establish the mixing time to be in between $\frac{n}{k \log k}\log n$ and $\frac{n}{k-1}\log n$.