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We investigate the percolation transition of aligned, overlapping,non-symmetrical shapes on lattices. Using the recently proposed lattice versionof excluded volume theory, we show that shape-asymmetry leads to someintriguing consequences regarding the percolation behavior of asymmetricshapes. We consider a prototypical asymmetric shape - rectangle - on a squarelattice and show that for rectangles of width unity (sticks), the percolationthreshold is a monotonically decreasing function of the stick length, whereas,for rectangles of width greater than two, it is a monotonically increasingfunction. Interestingly, for rectangles of width two, the percolation thresholdis independent of its length. The limiting case of the length of the rectanglesgoing to infinity shows that the limiting threshold value is finite and dependsupon the width of the rectangle. Unlike the case of symmetrical shapes likesquares, there seems to be no continuum percolation problem that corresponds tothis limit. We show that similar results hold for other asymmetric shapes andlattices. The critical properties of the aligned and overlapping rectangles areevaluated using Monte Carlo simulations. We find that the threshold valuesgiven by the lattice-excluded volume theory are in good agreement with thesimulation results, especially for larger rectangles. We verify the isotropy ofthe percolation threshold and also compare our results with models whererectangles of mixed orientation are allowed. Our simulation results show thatalignment increases the percolation threshold. The calculation of criticalexponents places the model in the standard percolation universality class. Ourresults show that shape-anisotropy of the aligned, overlapping percolatingunits has a marked influence on the percolation properties, especially when asubset of the dimensions of the percolation units are made to diverge.

Effect of shape asymmetry on percolation of aligned and overlapping objects on lattices