 at finite inter-layer Josephson coupling, and the approximation ignoring the compact nature of the variables becomes uncontrolled. As our analysis of one particular layered system shows, no SP exists in such a system despite the RG prediction: the system shows essentially the 3D behavior of the asymmetric XY model.

While it is not obvious to us what is wrong with treating Josephson coupling between 2D layers by RG in the original representation of fields, the dual formulation in terms of the closed loops gives a very important insight. Specifically, the SP means that as layer size $L\to \infty$, a suppression of the Josephson coupling between layers would require that the number of times elements of closed loops fluctuate between layers must scale slower than $L^2$ so that the density of such events is zero in the limit $L=\infty$. The loops statistics, however, is controlled by local energies of creating finite elements and the entropy due to 6 directions in 3D vs 4 along layers. Thus, as long as there is a finite energy to cross between neighboring layers, the entropy will lead to a finite density of crossings for large enough $L$. Similar argument can be applied to quantum wires in terms of the quantum to classical mapping where imaginary time is treated as an extra dimension. The dual approach and the argumentation along the line of the numerical algorithm [@WA], treating closed loops formation as a process of the worm head wondering around and eventually finding its tail, allowed us to expose the actual stages of the renormalization of the Josephson coupling: i) At small scales Josephson coupling is controlled by exponentially suppressed random and independent (in the asymptotic limit) events of crossings between layers. It can be viewed as an ideal gas of J-currents between the layers. This stage leads to the renormalized coupling, in general, represented by Eq.(\[GenM\]) with $M=1,2,3,...$. ii) If the number of layers $N_z$ increases, with $L$ being fixed, quasi 1D fluctuations further suppress the coupl