Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices. The process starts with an initial configuration, where all lattice sites are occupied and, obviously, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing straight rigid rods of length k (k-mers) from the surface. The central idea of this paper is based on finding the maximum concentration of occupied sites (minimum concentration of holes) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. The results, obtained for k ranging from 2 to 256, showed a nonmonotonic size k dependence for the critical concentration, which rapidly decreases for small particle sizes ( k 1 3 ⩽ ⩽ ). Then, it grows for k = 4, 5 and 6, goes through a maximum at k = 7, and finally decreases again and asymptotically converges towards a definite value for large values of k. Percolating and non-percolating phases extend to infinity in the space of the parameter k and, consequently, the model presents percolation transition in all ranges of said value. This finding contrasts with the results obtained in literature for a complementary problem, where straight rigid k-mers are randomly and irreversibly deposited on a square lattice, and the percolation transition only exists for values of k ranging between 1 and approximately × 1.2 10 4 . The breaking of particle-hole symmetry, a distinctive characteristic of the k-mers statistics, is the source of this asymmetric behavior.

The problem of inverse percolation by removing straight rigid rods from two-dimensional triangular lattices has been studied by using numerical simulations and finite-size scaling analysis. The process starts with an initial configuration, where all lattice sites are occupied and, consequently, the opposite sides of the lattice are connected by nearest-neighbor occupied sites. Then, the system is diluted by randomly removing straight rigid rods of a length k (k-mers) from the surface. Based on percolation theory, the maximum concentration of occupied sites (minimum concentration of holes) at which the connectivity disappears was obtained. This particular value of the concentration is named the inverse percolation threshold, and determines a well-defined geometrical (second order) phase transition in the system. The corresponding critical exponents were also calculated. The results, obtained for k ranging from 2 to 256, revealed that (i) the inverse percolation threshold exhibits nonmonotonic behavior as a function of the k-mer size: it grows from k = 1 to k = 10, goes through a maximum at k = 11, and finally decreases again and asymptotically converges towards a definite value for large values of k; (ii) the percolating and non-percolating phases extend to infinity in the space of the parameter k and, consequently, the model presents percolation transition in all the ranges of k; and (iii) the phase transition occurring in the system belongs to the standard random percolation universality class regardless of the value of k considered.

Irreversible adsorption of objects of different shapes and sizes on Euclidean, fractal and random lattices is studied. The adsorption process is modeled by using random sequential adsorption (RSA) algorithm. Objects are adsorbed on one-, two-, and three-dimensional Euclidean lattices, on Sierpinski carpets having dimension d between 1 and 2, and on Erdos-Renyi random graphs. The number of sites is M = L d for Euclidean and fractal lattices, where L is a characteristic length of the system. In the case of random graphs it does not exist such characteristic length, and the substrate can be characterized by a fixed set of M vertices (sites) and an average connectivity (or degree) g. The paper concentrates on measuring (1) the probability W L(M ) (θ) that a lattice composed of L d (M ) elements reaches a coverage θ, and (2) the exponent νj characterizing the so-called "jamming transition". The results obtained for Euclidean, fractal and random lattices indicate that the main quantities derived from the jamming probability W L(M ) (θ) behave asymptotically as M 1/2 . In the case of Euclidean and fractal lattices, where L and d can be defined, the asymptotic behavior can be written as M 1/2 = L d/2 = L 1/ν j , and νj = 2/d. arXiv:1907.02572v2 [cond-mat.stat-mech]

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices contaminated with non-conducting impurities. The presence of impurities provides a more realistic approach to the deposited monolayer, which usually presents inhomogeneities due to the irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc. The process starts with an initial configuration, where all lattice sites are occupied by an impurity (with a concentration ) or a conducting particle (with a concentration ). Then, the system is diluted by randomly removing linear k-mers (linear clusters of k consecutive conducting particles) from the lattice. The impurities remain fixed in its position and cannot be removed. The central idea of this paper is based on finding the maximum concentration of conducting sites (minimum concentration of empty sites) for which the connectivity disappears. This particular value of the concentration is called inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. On the other hand, the inverse jamming coverage is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbour sites of appropriate size. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 1 to 120. The obtained results show that the behaviour of the system is significantly affected by the presence of impurities. In addition, the nature of the jamming and percolation transitions was studied. In the first case, the corresponding spatial correlation length critical exponent was measured, being . This value coincides with previous calculations of this exponent for the standard random sequential adsorption of linear k-mers on square lattices. Critical exponents were also calculated for the percolation phase transition, showing that the universality class corresponding to ordinary percolation is preserved regardless the values of k and considered.