Spectral gap of a product of Markov processes

Question

For $m \in [N] \equiv \{1,\dots, N\}$, let $Q^{(m)}$ be the generator of a (well-behaved) continuous-time Markov process on a finite state space $[n_m]$. Write $J \equiv (j_1,\dots,j_N) \in \prod_m [n_m]$ with $j_m \in [n_m]$.

The composite Markov generator corresponding to running each of the $N$ processes independently has off-diagonal entries given by $Q^\otimes_{JJ'}=\lim_{\Delta \downarrow 0} \mathbb{P}(J \overset{\Delta}{\rightarrow} J')/\Delta$ for $J \ne J'$, where the probability of a transition from $J$ to $J'$ in a time interval of duration $\Delta$ is indicated. In this limit a.s. at most one of the component processes can execute a transition, so that the only nonzero off-diagonal terms are of the form $Q^\otimes_{JJ'} = Q^{(M)}_{j_M j'_M}$ with $j'_m = j_m$ for $m \ne M$.

What can be said about the spectral gap of $Q^\otimes$ or the mixing time of the underlying process?

Of particular interest is the case $Q^{(m)} \equiv Q$.

I've looked in the most obvious place (Levin, Peres, and Wilmer) and in some less obvious places, and I haven't seen this anywhere (perhaps I'm making a silly oversight). In the literature "product chains" usually mean something rather different than the discrete-time analogue of the above, and in general I expect that any terminology is overloaded. With that in mind, specific references would be most helpful.

Answer 1

The spectral gap is just the smallest of the spectral gaps of the component chains. The matrix $Q$ can be simply written as the sum of $Q^{(1)}\otimes I\otimes\ldots\otimes I$, $I\otimes Q^{(2)}\otimes I\otimes\cdots\otimes I$ etc.

If the matrix $Q^{(i)}$ has eigenvalues $(\lambda^{(i)}_j)_{j=1}^{n_i}$, then the matrix $Q$ has eigenvalues $\lambda^{(1)}_{i_1}+\lambda^{(2)}_{i_2}+\ldots+\lambda^{(N)}_{i_N}$ (with eigenvectors $e^{(1)}_{i_1}\otimes \cdots \otimes e^{(N)}_{i_N}$).

Answer 2

Hi Steve, Does section 12.4 of Levin, Peres, and Wilmer address your question? My first impression is that your problem corresponds to taking the weight $w_i$ in (12.19) of their book as $1/N$. Then the gap according to corollary 12.12 should be $\min_i w_i \gamma_i$ where $\gamma_i$ is the spectral gap of $Q^{(i)}$. Continuous time or discrete time shouldn't make a big difference to the spectral gap, as the former is just the exponentiation of the latter.