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.