We study diffusion of hard-core 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 ; an 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 below a critical threshold ρc=1 /(n +1 ) . We find that the ϕk, the density of 0-clusters (0 representing vacancies) of size 0 ≤k <n , vanish at the transition point along with activity density ρa. The steady state of these models can be written in matrix product form to obtain analytically the static exponents βk=n -k and ν =1 =η corresponding to each ϕk. We also show from numerical simulations that, starting from a natural condition, ϕk(t ) s decay as t-αk with αk=(n -k ) /2 even though other dynamic exponents νt=2 =z are independent of k ; this ensures the validity of scaling laws β =α νt and νt=z ν .