Percolation thresholds of randomly oriented patchy particles on Bethe lattices

In this paper, we study the percolation of patchy particles fully packed on the Bethe lattices. Each particle is a randomly oriented disk with n symmetrically distributed patches, and it is centered at a vertex of the lattice with coordination number z. The proportion [Formula: see text] of the patchy area on the disk is a control parameter. As [Formula: see text] increases, clusters of particles connected by patches grow and a spanning cluster appears at a threshold value [Formula: see text], i.e. the system percolates. We analytically calculated the threshold [Formula: see text] for [Formula: see text] and [Formula: see text]. For larger z, numerical simulations were used to estimate [Formula: see text]. By comparing these [Formula: see text] values, we find several rules. For example, the values of [Formula: see text] are same for certain models with fixed z and n satisfying specific relationships, and we also observed periodic rules of [Formula: see text] with fixed n and increasing z. The results can be useful references for further studying the connectivity of patchy particles on various lattices.