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Two Geometric Approaches To Study The Deconfinement Phase Transition in (3+1)-Dimensional $Z_2$ Gauge Theories

We have simulated $(3+1)-$dimensional finite temperature $Z_2$ gauge theory by using Metropolis algorithm. We aimed to observe the deconfinement phase transitions by using geometric methods. In order to do so we have proposed two different methods which can be applied to $3-$dimensional effective spin model consisting of Polyakov loop variables. The first method is based on studies of cluster structures of each configuration. For each temperature, configurations are obtained from a set of bond probability ($P$) values. At a certain probability, percolating clusters start to emerge. Unless the probability value coincides with the Coniglio-Klein probability value, the fluctuations are less than the actual fluctuations at the critical point. In this method the task is to identify the probability value which yields the highest peak in the diverging quantities on finite lattices. The second method uses the scaling function based on the surface renormalization which is of geometric origin. Since this function is a scaling function, the measurements done on different-size lattices yield the same value at the critical point, apart from the correction to scaling terms. The linearization of the scaling function around the critical point yields the critical point and the critical exponents.