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Percolation in a Multifractal

We build a multifractal object and use it as a support to study percolation. We identify some differences between percolation in a multifractal and in a regular lattice. We use many samples of finite size lattices and draw the histogram of percolating lattices against site occupation probability. Depending on a parameter characterizing the multifractal and the lattice size, the histogram can have two peaks. The percolation threshold for the multifractal is lower than for the square lattice. The percolation in the multifractal differs from the percolation in the regular lattice in two points. The first is related with the coordination number that changes along the multifractal. The second comes from the way the weight of each cell in the multifractal affects the percolation cluster. We compute the fractal dimension of the percolating cluster. Despite the differences, the percolation in a multifractal support is in the universality class of standard percolation.