Wenbo V. Li
Title: Spectral analysis of a class of non-reversible Markov Chains
Co-Author: Yuk J. Leung
Abstract:
We consider a class of non-reversible Markov Chains with transition matrix $A$ associated with
a discretization of Brownain Motion with jump boundary.
The discrete setting models a particle moving along integer
points ranging from $0$ to $n+1$ with probability $1/2$ to neighbors on interior points.
If the particle reaches either one of the end points $0$ or $n+1$,
it moves to the nearest neighbor with probability (at least) $1/2$
and the remaining probability to other points, not necessary the same for the two end points.
Thus the resulting matrix $A$ is non-symmetric and can be considered as a
perturbation of the well-known Jacobi matrix where the diagonal entries all zero
and both the upper and sub-diagonal entries are all 1/2.
We show by a zero-interlacing method
that the eigenvalues of $A$ are {\bf all} real and the
second eigenvalue in modulo is always negative (the first one is always $1$).
The asymptotic distribution of the eigenvalues as its size gets large is also considered.
Corrections:
Updated relevant references:
Last modified Oct. 10, 2010 by Wenbo V. Li,
wli@math.udel.edu