Multi-critical absorbing phase transition in a class of exactly solvable models

We study diffusion of hardcore particles on a one dimensional periodic lattice subjected to a constraint that the separation between any two consecutive particles does not increase beyond a fixed value $(n+1);$ initial separation larger than $(n+1)$ can however decrease. These models undergo an absorbing state phase transition when the conserved particle density of the system falls bellow a critical threshold $\rho_c= 1/(n+1).$ We find that $\phi_k$s, the density of $0$-clusters ($0$ representing vacancies) of size $0\le k<n,$ vanish at the transition point along with activity density $\rho_a$. The steady state of these models can be written in matrix product form to obtain analytically the static exponents $\beta_k= n-k,\nu=1=\eta$ corresponding to each $\phi_k$. We also show from numerical simulations that starting from a natural condition, $\phi_k(t)$s decay as $t^{-\alpha_k}$ with $\alpha_k= (n-k)/2$ even though other dynamic exponents $\nu_t=2=z$ are independent of $k$; this ensures the validity of scaling laws $\beta= \alpha \nu_t,$ $\nu_t = z \nu$.