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Additional material on bounds of $\ell^2$-spectral gap for discrete Markov chains with band transition matrices

We analyse the $\ell^2(π)$-convergence rate of irreducible and aperiodic Markov chains with $N$-band transition probability matrix $P$ and with invariant distribution $π$. This analysis is heavily based on: first the study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(π)})$ of $P\_{|\ell^2(π)}$ derived from Hennion's quasi-compactness criteria; second the connection between the spectral gap property (SG$\_2$) of $P$ on $\ell^2(π)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$) is shown to hold under the condition \[α\_0 := \sum\_{{m}=-N}^N \limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\, 1. \] Moreover $r\_{ess}(P\_{|\ell^2(π)}) \leq α\_0$. Simple conditions on asymptotic properties of $P$ and of its invariant probability distribution $π$ to ensure that $α\_0\textless{}1$ are given. In particular this allows us to obtain estimates of the $\ell^2(π)$-geometric convergence rate of random walks with bounded increments. The specific case of reversible $P$ is also addressed. Numerical bounds on the convergence rate can be provided via a truncation procedure. This is illustrated on the Metropolis-Hastings algorithm.