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

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.