Many-body contacts in fractal polymer chains and fBm trajectories

We calculate the probabilities that a trajectory of a fractional Brownian motion with arbitrary fractal dimension $d_f$ visits the same spot $n \ge 3$ times, at given moments $t_1, ..., t_n$, and obtain a determinant expression for these probabilities in terms of a displacement-displacement covariance matrix. Except for the standard Brownian trajectories with $d_f = 2$, the resulting many-body contact probabilities cannot be factorized into a product of single loop contributions. Within a Gaussian network model of a self-interacting polymer chain, which we suggested recently, the probabilities we calculate here can be interpreted as probabilities of multi-body contacts in a fractal polymer conformation with the same fractal dimension $d_f$. This Gaussian approach, which implies a mapping from fractional Brownian motion trajectories to polymer conformations, can be used as a semiquantitative model of polymer chains in topologically-stabilized conformations, e.g., in melts of unconcatenated rings or in the chromatin fiber, which is the material medium containing genetic information. The model presented here can be used, therefore, as a benchmark for interpretation of the data of many-body contacts in genomes, which we expect to be available soon in, e.g., Hi-C experiments.