# Spectral gap of continuous-time Markov chain on nonnegative integers: The geometric long indel length chain

Let $$S=\mathbb Z_{\ge0}$$ be the nonnegative integers, and let $$T=\mathbb R_{\ge0}$$ be the nonnegative real numbers. Next, let $$\gamma\in(0,1),r\in(0,1),$$ and let $$Q=(Q_{n,m})_{n,m\in S}$$ be such that for all integers $$n\ge0$$ and $$k\ge1,$$ $$Q(n,n+k)=(n+1)(\gamma r)^k,\qquad Q(n+k,n)=(n+1)r^k,\qquad Q(n,n)=-\sum_{m\in S:m\ne n}Q(n,m).$$

Now, is there a precise formula for the spectral gap, $$\operatorname{gap}(Q),$$ of $$Q$$? (Spectral gap in the sense of reversible continuous-time Markov chains, where we interpret $$Q$$ as an infinitesimal generator.)

My work: We can compare $$Q$$ with the matrix $$\tilde Q$$ defined by $$\tilde Q(n,n+1)=(n+1)\gamma r,\quad\tilde Q(n,n-1)=nr.$$ The spectral gap for $$\tilde Q$$ is known to be $$r(1-\gamma)$$ (Chen '04, Section 9.3, second example under heading "Examples 9.27"), and for all $$n,m\in S$$ such that $$n\ne m$$ we have $$\tilde Q(n,m)\le Q(n,m),$$ so that $$\operatorname{gap}(\tilde Q)\le\operatorname{gap}(Q).$$ Thus, $$r(1-\gamma)\le\operatorname{gap}(Q).$$ I do not know how to improve this bound or to obtain an upper bound.

(Note: This question is a simplification of a previous post of mine. In the notation of that post, we have $$\lambda_k=(\gamma r)^k$$ and $$\mu_k=r^k.$$)

Chen, Mu-Fa, From Markov chains to non-equilibrium particle systems., River Edge, NJ: World Scientific (ISBN 981-238-811-7/hbk; 978-981-256-245-6/ebook). xii, 597 p. (2004). ZBL1078.60003.

• crossposted, with a minor edit, from Math.SE – xFioraMstr18 Aug 10 '20 at 20:02