"[Consider] the function
zeta(z) = Trace((I-zP)^(-1))
where P is a stochastic matrix (i.e., a matrix of non-negative reals
with the entries in each row summing to 1).
Expanding zeta(z) as Trace(I) + z Trace(P) + z^2 Trace(P^2) + ...,
one can see that this is a sum of closed orbits in the state-space
of the Markov chain associated with the matrix P, where the weight
of a closed orbit of length n is z^n times the product of the
probabilities going around the orbit. So it's a sort of dynamical
zeta-function.
If one regularizes the trace at z=1 by subtracting off 1/(1-z), one
obtains a spectral invariant of P called Kemeny's constant, which
measures how "big" the Markov chain is. It has a nice probabilistic
interpretation; see math.dartmouth.edu/~doyle/docs/kc/kc.pdf.
So I was wondering if other aspects of zeta(z) have been studied
(e.g., if we differentiate it, does the regularized value at z=1
have a probabilistic interpretation? do the zeroes of the zeta(z)
obey any patterns?)"
email: prop 'at' jamespropp 'dot' org