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The fractal dimensions and the percolation exponents of the geometrical spin clusters of like sign at criticality, are obtained numerically for an Ising model with temperature-dependent annealed bond dilution, also known as the thermalized bond Ising model (TBIM), in two dimensions. For this purpose, a modified Wolff singlecluster Monte Carlo simulation is used to generate equilibrium spin configurations on square lattices in the critical region. A tie-breaking rule is employed to identify nonintersecting spin cluster boundaries along the edges of the dual lattice. The values obtained for the fractal dimensions of the spanning geometrical clusters D_c, and their interfaces D_I , are in perfect agreement with those reported for the standard twodimensional ferromagnetic Ising model. Furthermore, the variance of the winding
angles, results in a diffusivity $\kappa= 3$ for the two-dimensional thermalized bond Ising model, thus placing it in the universality class of the regular Ising model. A finite-size scaling analysis of the largest geometrical clusters, results in a reliable estimation of the critical percolation exponents for the geometrical clusters in the limit of an infinite lattice size. The percolation exponents thus obtained, are also found to be consistent with those reported for the regular Ising model. These consistencies are explained in terms of the Fisher renormalization relations, which express the thermodynamic critical exponents of systems with annealed bond dilution in terms of those of the regular model system
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