Comparing limit profiles of reversible Markov chains (Q6545181)

The article introduces a technique for comparing the limit profile of two Markov chains under certain conditions. More precisely, let \(X\) and \(X'\) be an ergodic Markov chain on a finite state space \(\Omega\), with implicit parameter \(n\), with the same equilibrium distribution \(\pi\). Assume that the transition matrices corresponding to \(X\) and \(X'\) commute. Let \(d_\mathsf{TV}(t)\) resp. \(d'_\mathsf{TV}(t)\) denote the total-variation distance between the time-\(t\) law of \(X\) resp. \(X'\) and \(\pi\). Suppose that there exist \(t_0\), \(t'_0\), \(w\) and \(w'\), all implicit functions of \(n\), such that\N\[\N\Phi(\alpha) := \lim_{n\to\infty} d_\mathsf{TV}(t_0 + \alpha w) \ \ \ \text{and} \ \ \ \Phi'(\alpha) := \lim_{n\to\infty} d'_\mathsf{TV}(t'_0 + \alpha w')\N\]\Nboth exist for all \(\alpha \in \mathbb R\); the function \(\Phi\) resp. \(\Phi'\) is the \textit{limit profile} of \(X\) resp. \(X'\). Theory is developed to compare \(\Phi\) and \(\Phi'\) in terms of the eigenvalues, see Lemma 1.4.\N\NThe author points out that \textit{any} of the usual random-transpositions shuffles, generated by the set of all transpositions in a symmetric group \(S_n\), commutes with any other symmetric random walk \(S_n\). This allows her to use the limit profile for the random-transposition shuffle, found recently by \textit{L. Teyssier} [Ann. Probab. 48, No. 5, 2323--2343 (2020; Zbl 1468.60091)], to bound the limit profile of other shuffles. Particularly, an analogous Poissonian limit for the star-transpositions shuffle at time \(n (\log n + c)\) is established as Theorem~1.3:\N\[\N\lim_{n\to\infty} d_\mathsf{TV}\bigl( n (\log n + c) \bigr) = d_\mathsf{TV}\bigl( \operatorname{Pois}(1 + e^{-c}), \operatorname{Pois}(1) \bigr).\N\]