The longest excursion of a random interacting polymer

TL;DR

This paper analyzes the longest excursion length of a random interacting polymer, showing it follows a Gumbel distribution and is typically of order log N, using extreme value and renewal theories.

Contribution

It derives a limit law for the longest excursion of an interacting polymer, revealing its Gumbel distribution and order of magnitude.

Findings

Longest excursion length scales as log N

Longest excursion length converges to Gumbel distribution

Provides a law of large numbers for excursion length

Abstract

We consider a random $N$-step polymer under the influence of an attractive interaction with the origin and derive a limit law -- after suitable shifting and norming -- for the length of the longest excursion towards the Gumbel distribution. The embodied law of large numbers in particular implies that the longest excursion is of order $\log N$ long. The main tools are taken from extreme value theory and renewal theory.