We characterize the law of the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice. The proof is via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion. It is based on a mapping which can be regarded as a geometric variant of the RSK correspondence.
"Directed polymers and the quantum Toda lattice." Ann. Probab. 40 (2) 437 - 458, March 2012. https://doi.org/10.1214/10-AOP632