Persi Diaconis (Stanford University)

Tri-diagonal matrices and alternating permutations

The space of double stochastic tri-diagonal matrices has interesting combinatorics:it has a Fibbonacci number of extreme points,The Eulerian number for volume...As a compact convex subset, it makes sense to talk about such a matrix 'picked at random'. For a probabalist, this amounts to looking at a typical birth and death chain. One can ask about the distribution of the eigenvalues and entries and rate of convergence to stationarity. All of this is closely connected with the well studies space of alternating permutations on n letters.The analysis leads to classical sets of orthogonal polynomials and shows that 'most symmetric birth and death chains do not have a sharp cutoff in their rate of convergence to stationarity'. This is joint work with Phillip Matchup-Wood.