Random Polymers and Generalized Urn Processes

We describe a microcanonical approach for polymer models that combines atmospheric methods with urn theory. We show that Large Deviation Properties of urn models can provide quite deep mathematical insight by analyzing the Random Walk Range problem in $\mathbb{Z}^{d}$. We also provide a new mean field theory for the Range Problem that is exactly solvable by analogy with the Bagchi-Pal urn model.