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This work studies the jamming and percolation of parallel squares in a single-cluster growth model. The Leath-Alexandrowicz method was used to grow a cluster from an active seed site. The sites of a square lattice were occupied by addition of the equal size $k times k$ squares (E-problem) or a mixture of $k times k$ and $m times m$ ($m leqslant k$) squares (M-problem). The larger $k times k$ squares were assumed to be active (conductive) and the smaller $m times m$ squares were assumed to be blocked (non-conductive). For equal size $k times k$ squares (E-problem) the value of $p_j = 0.638 pm 0.001$ was obtained for the jamming concentration in the limit of $krightarrowinfty$. This value was noticeably larger than that previously reported for a random sequential adsorption model, $p_j = 0.564 pm 0.002$. It was observed that the value of percolation threshold $p_{mathrm{c}}$ (i.e., the ratio of the area of active $k times k$ squares and the total area of $k times k$ squares in the percolation point) increased with an increase of $k$. For mixture of $k times k$ and $m times m$ squares (M-problem), the value of $p_{mathrm{c}}$ noticeably increased with an increase of $k$ at a fixed value of $m$ and approached 1 at $kgeqslant 10m$. This reflects that percolation of larger active squares in M-problem can be effectively suppressed in the presence of smaller blocked squares.
We investigate kinetically constrained models of glassy transitions, and determine which model characteristics are crucial in allowing a rigorous proof that such models have discontinuous transitions with faster than power law diverging length and ti
We perform Monte Carlo simulations to determine the average excluded area $<A_{ex}>$ of randomly oriented squares, randomly oriented widthless sticks and aligned squares in two dimensions. We find significant differences between our results for rando
We study the glass and jamming transition of finite-dimensional models of simple liquids: hard- spheres, harmonic spheres and more generally bounded pair potentials that modelize frictionless spheres in interaction. At finite temperature, we study th
We present a novel mechanism for the anomalous behaviour of the specific heat in low-temperature amorphous solids. The analytic solution of a mean-field model belonging to the same universality class as high-dimensional glasses, the spherical percept
Neutron scattering experiments at the magnetic vacancy percolation threshold concentration, x_v, using the random-field Ising crystal Fe(0.76)Zn(0.24)F2, show stability of the transition to long-range order up to fields H=6.5 T. The observation of th