Longest increasing subsequences of random colored permutations

We compute the limit distribution for (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In two--colored case our method provides a different proof of a similar result by Tracy and Widom about longest increasing subsequences of signed permutations (math.CO/9811154). Our main idea is to reduce the `colored' problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.Comment: AMSTeX, 11 page

Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures. One application is a simple exact sampling algorithm for q^{volume}-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand-Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.Comment: 22 page

Loop-free Markov chains as determinantal point processes

We show that any loop-free Markov chain on a discrete space can be viewed as a determinantal point process. As an application, we prove central limit theorems for the number of particles in a window for renewal processes and Markov renewal processes with Bernoulli noise.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP115 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org