Gross-Ooguri Phase Transition at Zero and Finite Temperature: Two Circular Wilson Loop Case

In the context of $AdS\mathrm{/}CFT$ correspondence the two Wilson loop correlator is examined at both zero and finite temperatures. On the basis of an entirely analytical approach we have found for Nambu-Goto strings the functional relation $d{S}_c^{(Reg)}\mathrm{/}dL=2\pi k$ between Euclidean action $S_c$ and loop separation $L$ with integration constant $k$, which corresponds to the analogous formula for point-particles. The physical implications of this relation are explored in particular for the Gross-Ooguri phase transition at finite temperature.