Directed Polymers and the Quantum Toda Lattice By

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. 1. Introduction. Let B 1 (t), B 2 (t),. .. , B N (t), t ≥ 0, be a collection of independent standard one-dimensional Brownian motions and write B i (s, t) = B i (t) − B i (s) for s ≤ t. Let β ∈ R, t ≥ 0, and consider the random variable