Poisson limit of bumping routes in the Robinson–Schensted correspondence

Joint work with Dan Romik, Łukasz Maślanka, Mikołaj Marciniak.
We are interested in asymptotic questions related to Robinson–Schensted–Knuth algorithm applied to a random input and Plancherel measure on the set of infinite standard Young tableaux. One of such questions concerns the shape of the bumping route when a specified number is inserted into a large (or infinite) Plancherel-distributed tableau; somewhat surprisingly this problem turns out to be equivalent to the question (stated by Vershik in 2020) about the time-evolution of the position of a specified number in the insertion tableau as more and more numbers are inserted.
We focus on the direct vicinity of the $y$-axis, for example we are interested in the time it takes for the bumping route / box to reach the first column of the tableau. Asymptotically, the trajectory turns out to converge in distribution to an explicit random process.
Further reading: https://arxiv.org/abs/2005.14397