Ordering and percolation transitions for hard squares:: Equilibrium versus nonequilibrium models for adsorbed layers with c(2x2) superlattice ordering

We study the critical behavior of models for adsorbed layers in which particles reside on a square lattice and have infinite nearest-neighbor repulsions. Such particles are often described as "hard squares." We consider both the equilibrium hard-square model and a nonequilibrium model. The latter involves dimer adsorption onto diagonally adjacent sites, and the desorption and possible hopping of adsorbed monomer particles (where neither adsorption nor hopping can create adjacent pairs of occupied sites). In the limit or high monomer mobility, one recovers the equilibrium model. Both models exhibit a continuous symmetry breaking transition in the Ising universality class, and also a percolation transition for c(2x2) clusters of particles connected with diagonal bonds. For the equilibrium model, extensive Monte Carlo simulations show that the two transitions coincide, supporting the claim of Hu and Mak. We also determine percolation exponents for c(2x2) clusters and vacancy clusters, and consider a correlated site-bond percolation problem which elucidates conditions for coincidence of symmetry-breaking and percolation. In contrast, for the nonequilibrium model with immobile adsorbed monomers, there is a gap between the symmetry-breaking and percolation transitions, and the random percolation universality class applies. Finally, we examine the crossover behavior with increasing mobility of adsorbed monomers.