Coupled map lattice with pairwise local interactions is a well-studied system. However, in several situations, such as neuronal or social networks, multi-site interactions are possible. In this work, we study the coupled Gauss map in one dimension with 2-site, 3-site, 4-site and 5-site interaction. This coupling cannot be decomposed in pairwise interactions. We coarse-grain the variable values by labeling the sites above x^ as up spin (+1) and the rest as down spin (-1) where x^ is the fixed point. We define flip rate F (t) as the fraction of sites i such that s₈ (t-1) s₈ (t) and persistence P (t) as the fraction of sites i such that s₈ (t') =s₈ (0) for all t' t. The dynamic phase transitions to a synchronized state is studied above quantifiers. For 3 and 5 sites interaction, we find that at the critical point, F (t) t^- with =0. 159 and P (t) t^- with =1. 5. They match the directed percolation (DP) class. Finite-size and off-critical scaling is consistent with DP class. For 2 and 4 site interactions, the exponent and behavior of P (t) at critical point changes. Furthermore, we observe logarithmic oscillations over and above power-law decay at the critical point for 4-site coupling. Thus multi-site interactions can lead to new universality class (es).
Warambhe et al. (2025) studied this question.