We will discuss a new approach to the analysis of the global behavior of stochastic discrete particle systems. This approach links the asymptotics of these systems with properties of certain observables related to the Schur symmetric functions.As applications of this method, we prove the Law of Large Numbers and the Central Limit Theorem for various models of random lozenge and domino tilings, non-intersecting random walks, and decompositions of tensor products of representations of unitary groups.Based on joint works with V. Gorin and A. Knizel.