Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Additional material on bounds of $\ell^2$-spectral gap for discrete
  Markov chains with band transition matrices   We analyse the $\ell^2(\pi)$-convergence rate of irreducible and aperiodic
Markov chains with $N$-band transition probability matrix $P$ and with
invariant distribution $\pi$. This analysis is heavily based on: first the
study of the essential spectral radius $r\_{ess}(P\_{|\ell^2(\pi)})$ of
$P\_{|\ell^2(\pi)}$ derived from Hennion's quasi-compactness criteria; second
the connection between the spectral gap property (SG$\_2$) of $P$ on
$\ell^2(\pi)$ and the $V$-geometric ergodicity of $P$. Specifically, (SG$\_2$)
is shown to hold under the condition \[\alpha\_0 := \sum\_{{m}=-N}^N
\limsup\_{i\rightarrow +\infty} \sqrt{P(i,i+{m})\, P^*(i+{m},i)}\ \textless{}\,
1. \] Moreover $r\_{ess}(P\_{|\ell^2(\pi)}) \leq \alpha\_0$. Simple conditions
on asymptotic properties of $P$ and of its invariant probability distribution
$\pi$ to ensure that $\alpha\_0\textless{}1$ are given. In particular this
allows us to obtain estimates of the $\ell^2(\pi)$-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.

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The categories of this paper are: patt-sol
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph