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Continuity for Limit Profiles of Reversible Markov Chains

Evita Nestoridi

Abstract

We prove that the limit profile of a sequence of reversible Markov chains exhibiting total variation cutoff is a continuous function, under a computable condition involving the spectrum of the transition matrix and the cutoff window.

Continuity for Limit Profiles of Reversible Markov Chains

Abstract

We prove that the limit profile of a sequence of reversible Markov chains exhibiting total variation cutoff is a continuous function, under a computable condition involving the spectrum of the transition matrix and the cutoff window.

Paper Structure

This paper contains 13 sections, 4 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['main']}
  4. The discrete time limit profile
  5. Transitive chains
  6. Examples
  7. Lazy random walk on the hypercube
  8. Random Walks on Ramanujan Graphs
  9. Random--to--random
  10. Random and star transpositions, and $k$--cycles
  11. The Bernoulli--Laplace chain
  12. Biased Card Shuffling and ASEP
  13. Acknowledgements

Key Result

Theorem 3

Let $P_n$ be the transition matrix of a reversible Markov chain on $X_n$ that exhibits cutoff at $t_n$ with window $w_n$ and satisfies cond. Assume that $\Phi_x$, the total variation limit profile with respect to the sequences $(t_n)$ and $(w_n)$ at $x$, exists. Then $\Phi_x$ is continuous for every

Theorems & Definitions (9)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • proof : Proof of Theorem \ref{['main']}
  • Theorem 5
  • proof
  • Theorem 6
  • Remark 7