ERDOS-RENYI LAWS FOR GIBBS MEASURES

Can one detect a phase transition from a single, large sample of a Gibbs measure? What information does one get on the other Gibbs distributions with the same potential? We approach these questions via Erdos-Renyi laws. In particular we prove almost-sure limit theorems for sets of empirical distributions of sub-samples of the given one: for suitable sub-samples size this set converges to the set of stationary Gibbs measures. Moreover we formulate Erdos-Renyi laws for general families of random variables with suitable large deviation principles.