Random Polymers and Generalized Urn Processes

TL;DR

This paper introduces a novel microcanonical approach combining atmospheric methods and urn theory to analyze polymer models, providing new insights into large deviation properties and a solvable mean field theory for the Range Problem.

Contribution

It presents a new microcanonical framework for polymer models, linking urn theory with atmospheric methods, and offers a solvable mean field theory for the Range Problem.

Findings

Large deviation properties of urn models yield deep mathematical insights.

A new exactly solvable mean field theory for the Range Problem is developed.

The approach connects polymer models with generalized urn processes.

Abstract

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.