Wright-Fisher model with negative mutation rates
ABSTRACT:
An n-leaf cladogram is an unrooted tree with n leaves and every internal
vertex of degree three. David Aldous introduced a natural Markov chain on
the space of Cladograms that consists of picking a leaf at random and
randomly re-attaching it to one of the remaining edges. As n grows to
infinity this diffusion, suitably rescaled in time, is conjectured to
converge to a diffusion on the space of continuum trees. We talk about the
local structure of the limiting diffusion seen from the point of view of
finitely many branch points. Locally it is a finite dimensional diffusion
with state space in the unit simplex and is not reversible. However we
show how it is related to another famous Markov chain model, the
Wright-Fisher model, which consists of sampling balls of different colors
from one urn to fill another. The key seems to be interpreting these
diffusions as functions of Galton-Watson trees with immigration. As a
benefit of working with diffusions explicit calculations are easy to
derive.
BIO:
Soumik Pal did his undergraduate at the Indian Statistical Institute in
Kolkata, and got his Ph.D. at the statistics department of Columbia
University, under the supervision of I. Karatzas in 2006. After a postdoc
at Cornell, he became a professor in the math department at UW in 2008.