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PAC

Information Center for Mathematical Science

PAC

On Dirichlet eigenvectors for neutral two-dimensional Markov chains
Author Persi Diaconis (Stanford University)
Homepage Url http://www-stat.stanford.edu/~cgates/PERSI/year.html
Coauthors Nicolas Champagnat, Laurent Miclo
Abstract We consider a general class of discrete, two-dimensional Markov chains modeling the dynamics of a population with two types, without mutation or immigration, and neutral in the sense that type has no influence on each individual’s birth or death parameters. We prove that all the eigenvectors of the corresponding transition matrix or infinitesimal generator  can be expressed as the product of explicit “universal” polynomials of two variables, depending on each type’s size but not on the specific transitions of the dynamics, and functions depending only on the total population size. We also prove that all the Dirichlet eigenvectors of  on subdomains of the form f(i; j) 2 N2 : i+j  Ng for some N  2 have the same decomposition. We then deduce that all the corresponding Dirichlet eigenvalues are ordered in a specific way and each of them is related to the greatest eigenvalue associated to eigenvectors admitting one specific “universal” polynomial as factor. As an application, we study the quasistationary behavior of finite, two-dimensional Markov chains such that 0 is an absorbing state for each component of the process. In particular, we prove that coexistence is never possible conditionally on non-extinction in a population close to neutrality.
Abstract Url http://www-stat.stanford.edu/~cgates/PERSI/papers/dirichlet-Jour-Prob.pdf