Abstract
This paper studies a number of matrix models of size n and the associated
Markov chains for the eigenvalues of the models for consecutive n's. They are
consecutive principal minors for two of the models, GUE with external source
and the multiple Laguerre matrix model, and merely properly defined consecutive
matrices for the third one, the Jacobi-Pineiro model; nevertheless the
eigenvalues of the consecutive models all interlace. We show: (i) For each of
those finite models, we give the transition probability of the associated
Markov chain and the joint distribution of the entire interlacing set of
eigenvalues; we show this is a determinantal point process whose extended
kernels share many common features. (ii) To each of these models and their set
of eigenvalues, we associate a last-passage percolation model, either finite
percolation or percolation along an infinite strip of finite width, yielding a
precise relationship between the last passage times and the eigenvalues. (iii)
Finally it is shown that for appropriate choices of exponential distribution on
the percolation, with very small means, the rescaled last passage times lead to
the Pearcey process; this should connect the Pearcey statistics with random
directed polymers.