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 $0le k<n,$ 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 $beta_k= n-k, u=1=eta$ 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 $ u_t=2=z$ are independent of $k$; this ensures the validity of scaling laws $beta= alpha u_t,$ $ u_t = z u$.