| Abstract: |
| Beginning from a directed polymer in a Gaussian random environment, we derive a hierarchical PDE system from the annealed endpoint density. A natural approximation of this in the annealed setting yields a simple non-local reaction diffusion equation. An open conjecture in probability that has been checked in certain exactly solvable cases is that, in one dimension, this directed polymer is in the KPZ universality class, which exhibits scaling in space with a 2/3 power law in time. We show that our approximate model exhibits non-trivial behavior on the same spatial scales (that is, a 2/3 power law in time) and identify the limiting distribution by connecting to the Fisher-KPP equation via a careful rescaling and change-of-coordinates. |
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