Generating equilateral random polygons in confinement
Yuanan Diao, Claus Ernst, Anthony Montemayor, Uta Ziegler · Journal of Physics A Mathematical and Theoretical · 2011
The paper describes an algorithm that generates equilateral random polygons confined in a sphere of fixed radius. Here, an equilateral random polygon is generated sequentially starting and ending at the origin by reversing the order (vertex by vertex) of equilateral random walks with fixed end points. Once a vertex X k +1 is generated, the next vertex X k is generated subject to two conditions: (1) X 0 = O , X 1 , ... X k , X k +1 form an equilateral random walk with fixed end points X 0 = O and X k +1 , and (2) X k is within the confining sphere. In the abstract of the paper, we mistakenly stated that this algorithm generates an equilateral random polygon confined in a sphere according to its probability distribution. It does not. Although the second condition ensures that the polygon generated is within the confining sphere, the condition to capture the theoretical probability distribution is: (2') X j is within the confining sphere for 1 ≤ j ≤ k . The authors will soon submit another paper on the generating method using conditions (1) and (2').