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

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, L 2 π . Specifically, we study analytically and via off-lattice Monte-Carlo simulations relative fluctuations of chain monomers in an ensemble of Brownian loops. We have shown that these fluctuations in the regime with fixed large gyration radius are Gaussian with the critical exponent γ = 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 γ = 1 3 typical for the Kardar–Parisi–Zhang universality class.

Stretching of a fractal polymer around a disc reveals KPZ-like statistics

“…As one can infer from this plot, the distributions remarkably collapse on the squared Airy function, which also describes the one-point distribution of fluctuation in the Ferrari-Spohn process, i.e. constrained (1+1)D random paths above the semicircle [10,14,16].…”

“…We should emphasize that the above mentioned similarity, however attractive, is not complete. Although valid on averages, it cannot be extended on distributions: the partition functions of a polymer in quenched Gaussian disorder and fluctuations of the topmost vicious walks have the Tracy-Widom distribution, while the constrained random walk above the boundary is given by squared Airy function [14,16]. Here we report the equivalent squared Airy PDF of fluctuations in the stretched polymer problem at various degrees of stretching (Fig.…”

“…Although the latter result is exact, addressing the limit of strongly stretched chain based on this expression remains a steep challenge. A significant progress in this direction was recently achieved by B.Meerson and N.Smith [10,11], some related problems were also examined by some of us [12][13][14].…”

Geometrical optics of large deviations of Brownian motion in inhomogeneous media

“…Large deviations of Brownian motion have recently attracted much attention from statistical physicists [1][2][3][4][5][6][7][8][9][10][11][12][13]. One setting where large deviations arise naturally is stochastic searching of a distant target by multiple random searchers, such as receptors searching for a reaction site on the surface of a living cell [14], or sperm cells searching for an oocyte [15].…”