Fixman problem revisited: When fluctuations of inflated ideal polymer loop are non-Gaussian?

TL;DR

This paper investigates the fluctuation statistics of an ideal polymer loop near its inflated state, revealing Gaussian fluctuations in certain regimes and non-Gaussian behavior with KPZ universality when an obstacle is introduced.

Contribution

It analytically and computationally demonstrates the transition from Gaussian to non-Gaussian fluctuations in polymer loops under specific conditions, highlighting a KPZ universality class behavior.

Findings

Fluctuations are Gaussian with exponent 1/2 when the gyration radius is close to the maximum.

Introducing an impenetrable disc causes fluctuations to become non-Gaussian with exponent 1/3.

The non-Gaussian regime aligns with KPZ universality class characteristics.

Abstract

We consider statistics of a planar ideal polymer loop of length $L$ in a large deviation regime, when a gyration radius, $R_g$, is slightly less than the radius of a fully inflated ring, $\frac{L}{2\pi}$. Specifically, we study analytically and via off-lattice Monte-Carlo simulations relative fluctuations of chain monomers in ensemble of Brownian loops. We have shown that these fluctuations in the regime with fixed large gyration radius are Gaussian with the critical exponent $\gamma = \frac{1}{2}$. However, if we insert inside the inflated loop the impenetrable disc of radius $R_d=R_g$, the fluctuations become non-Gaussian with the critical exponent $\gamma=\frac{1}{3}$ typical for the Kardar-Parisi-Zhang universality class.