We relate a continuous-time version of the geometric RSK correspondence to the Toda lattice, in a way which can be viewed as a semi-classical limit of a recent result by the author which relates the continuous-time geometric RSK mapping, with Brownian motion as input, to the quantum Toda lattice.
"Geometric RSK and the Toda lattice." Illinois J. Math. 57 (3) 883 - 918, Fall 2013. https://doi.org/10.1215/ijm/1415023516