There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane partitions of shape $n\times n$. See for instance the exposition given here: <http://www-users.math.umn.edu/~shopkins/docs/rsk.pdf>.


Now, since reverse plane partitions of shape $n\times n$ are a subset of the $\mathbb{N}$-matrices, it is possible to consider iterates of this map. Note that the total "mass" in the system is growing unboundedly: we have that $|\widehat{A}| := \sum\_{(i,j) \in [n]\times[n]} \widehat{a}\_{i,j} = \sum\_{(i,j) \in [n] \times [n]} a\_{i,j} h(i,j)$, where $h(i,j)$ is the hook-length of the box in position $(i,j)$. So probably one wants to renormalize in some way in order to consider asymptotic behaviour. My question is just whether there is any existing literature on the dynamics of RSK.



Dynamics of RSK
The paper "Minuscule reverse plane partitions via quiver representations" by Garver, Patrias, and Thomas (<https://arxiv.org/abs/1812.08345>) does a good job explaining, from the perspective of quiver representations, how the piecewise-linear RSK map is related to another invertible piecewise-linear operation called "rowmotion" (also "promotion"), which does have very interesting dynamics. I think this is the best answer to the vague circle of ideas I was prodding at.


