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Elastic Lattice Polymers

We study a model of &#34;elastic&#34; lattice polymer in which a fixed number of monomers $m$ is hosted by a self-avoiding walk with fluctuating length $l$. We show that the stored length density $ρ_m = 1 - <l>/m$ scales asymptotically for large $m$ as $ρ_m=ρ_\infty(1-θ/m + ...)$, where $θ$ is the polymer entropic exponent, so that $θ$ can be determined from the analysis of $ρ_m$. We perform simulations for elastic lattice polymer loops with various sizes and knots, in which we measure $ρ_m$. The resulting estimates support the hypothesis that the exponent $θ$ is determined only by the number of prime knots and not by their type. However, if knots are present, we observe strong corrections to scaling, which help to understand how an entropic competition between knots is affected by the finite length of the chain.