Asymptotic distribution of the size of a $d$-dimensional model of an open string

In this paper, we propose a statistical description of the configuration of a $d$-dimensional model of an open string that avoids self-intersection. For the distribution of the distance between the ends of the string we have obtained an exact integral equation similar to the well-known Dyson equation in quantum field theory. The resulting equation is invariant under continuous transformations of the renormalization group, which allows one to use the renormalization group method to establish the desired asymptotic distribution.