Is directed percolation class for synchronization transition robust with multi-site interactions?

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 $x^{\star}$ as up spin (+1) and the rest as down spin (-1) where $x^{\star}$ is the fixed point. We define flip rate $F(t)$ as the fraction of sites $i$ such that $s_{i}(t-1) \neq s_{i}(t)$ and persistence $P(t)$ as the fraction of sites $i$ such that $s_{i}(t')=s_{i}(0)$ for all $t' \le t$. 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, $F(t) \sim t^{-\delta}$ with $\delta=0.159$ and $P(t) \sim t^{-\theta}$ with $\theta=1.5$. 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 $\delta$ 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).