  In recent work on nonequilibrium statistical physics, a certain Markovian
exclusion model called an asymmetric annihilation process was studied by Ayyer
and Mallick. In it they gave a precise conjecture for the eigenvalues (along
with the multiplicities) of the transition matrix. They further conjectured
that to each eigenvalue, there corresponds only one eigenvector. We prove the
first of these conjectures by generalizing the original Markov matrix by
introducing extra parameters, explicitly calculating its eigenvalues, and
showing that the new matrix reduces to the original one by a suitable
specialization. In addition, we outline a derivation of the partition function
in the generalized model, which also reduces to the one obtained by Ayyer and
Mallick in the original model.
