In this talk we discuss closed self adjoint extensions of the Laplacian and fractional Laplacian on $L^2$ of Euclidean space minus the origin. In many cases there is a one parameter family of these operators that behave like the original operator plus a potential at the origin. Using these operators, we can construct polymer measures which exhibit interesting phase transitions from an extended state to a bound state where the pinning at the origin due to the potential takes over. The talk is based on joint works with Koralov, Molchanov, Squartini and Vainberg.