Linking of uniform random polygons in confined spaces

In this paper, we study the topological entanglement of uniform random polygons in a confined space. We derive the formula for the mean squared linking number of such polygons. For a fixed simple closed curve in the confined space, we rigorously show that the linking probability between this curve and a uniform random polygon of n vertices is at least 1-O(1/root n). Our numerical study also indicates that the linking probability between two uniform random polygons (in a confined space), of m and n vertices respectively, is bounded below by 1-O(1/root mn). In particular, the linking probability between two uniform random polygons, both of n vertices, is bounded below by 1-O(1/n).