In this talk, I will introduce the completely packed loop models (CPL) as a “transition diagram” between two link patterns. We will see a connection between the weighted counting of such transition diagrams and the transfer matrix of six vertex models with twisted periodic boundary conditions. I will also present a Markov chain model on linking patterns by sampling semi-infinite cylindrical tilings of independent Bernoulli CPL plaquettes. If time permits, I will discuss the Razumov-Stroganov conjecture that attempts to give an explicit combinatorial description of this probability distribution via six vertex models with domain wall boundary conditions.