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 rho(c) = 1/( n + 1). We find that the phi(k), the density of 0-clusters (0 representing vacancies) of size 0 <= k < n, vanish at the transition point along with activity density rho(a). The steady state of these models can be written inmatrix product form to obtain analytically the static exponents beta(k) = n - k and 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) and nu(t) = z nu.
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