Canonical ensemble of particles in a self-avoiding random walk

We consider an ensemble of particles not interacting with each other and randomly walking in the $d$-dimensional Euclidean space $\mathbb R^d$. The individual moves of each particle are governed by the same distribution, but after the completion of each such move of a particle, its position in the medium is "marked" as a region in the form of a ball of diameter $r_0$, which is not available for subsequent visits by this particle. As a result, we obtain the corresponding ensemble in $\mathbb R^d$ of marked trajectories in each of which the distance between the centers of any pair of these balls is greater than $r_0$. We describe a method for computing the asymptotic form of the probability density $W_n(\mathbf r)$ of the distance $r$ between the centers of the initial and final balls of a trajectory consisting of $n$ individual moves of a particle of the ensemble. The number $n$, the trajectory modulus, is a random variable in this model in addition to the distance $r$. This makes it necessary to determine the distribution of $n$, for which we use the canonical distribution obtained from the most probable distribution of particles in the ensemble over the moduli of their trajectories. Averaging the density $W_n(\mathbf r)$ over the canonical distribution of the modulus $n$ allows finding the asymptotic behavior of the probability density of the distance $r$ between the ends of the paths of the canonical ensemble of particles in a self-avoiding random walk in $\mathbb R^d$ for $2\le d<4$.