Staircase tableaux are a relatively new combinatorial structure introduced by Sylvie Corteel and Lauren Williams in the context of the asymmetric exclusion process (ASEP) and Askey--Wilson polynomials. Since then, their purely combinatorial properties have gained considerable interest.

I will consider a general model in which symbols that appear in tableaux may have arbitrary positive weights. I will mention some results concerning the limiting laws for the number of appearances of symbols in a random staircase tableau.

One advantage of this generality is that one may let the weights approach extreme values of zero or infinity which leads to tableaux with special properties. Furthermore, this generality allows us to make first steps in the analysis of the structure of a random staircase tableau and I will present some results in this direction.

The talk is based on a joint work with Svante Janson, Uppsala University (http://arxiv.org/abs/1212.5498)