 
Summary: Scaling behavior of random knots
Akos Dobay*, Jacques Dubochet*, Kenneth Millett
, PierreEdouard Sottas
, and Andrzej Stasiak*§
*Laboratory of Ultrastructural Analysis, University of Lausanne, 1015 Lausanne, Switzerland; Department of Mathematics, University of California,
Santa Barbara, CA 93106; and Center for Neuromimetic Systems, Swiss Federal Institute of Technology, 1015 Lausanne, Switzerland
Communicated by Sergei P. Novikov, University of Maryland, College Park, MD, February 14, 2003 (received for review October 20, 2002)
Using numerical simulations we investigate how overall dimen
sions of random knots scale with their length. We demonstrate
that when closed nonselfavoiding random trajectories are di
vided into groups consisting of individual knot types, then each
such group shows the scaling exponent of 0.588 that is typical for
selfavoiding walks. However, when all generated knots are
grouped together, their scaling exponent becomes equal to 0.5 (as
in nonselfavoiding random walks). We explain here this apparent
paradox. We introduce the notion of the equilibrium length of
individual types of knots and show its correlation with the length
of ideal geometric representations of knots. We also demonstrate
that overall dimensions of random knots with a given chain length
follow the same order as dimensions of ideal geometric represen
