In this work, we have proposed a new geometrical method for calculating the critical temperature and critical exponents by introducing a set of bond breaking probability values. The probability value Pc corresponding to the Coniglio–Klein probability for the transition temperature is obtained among this set of trial probabilities. Critical temperature, thermal and magnetic exponents are presented for d = 2 and d = 3, q = 2 Potts model and for the application of the method to the system with first order phase transition, q = 7 Potts model on different size lattices are employed. The advantage of this method can be that the bond breaking probability can be applied, where the clusters are defined on a set of dynamic variables, which are different from the dynamic quantities of the actual Hamiltonian or the action of the full system. An immediate application can be to use the method on finite temperature lattice gauge theories.
The scaling behaviors of the percolation cumulant and the surface renormalization are studied on q = 2 and 7 state Potts models. The results show that the scaling functions can be safely used to determine infinite lattice transition points and the thermal and magnetic exponents indicating that these functions have very small correction to scaling contributions.
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