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Mixing on the cycle with constant size perturbation

Shi Feng, Balázs Gerencsér

Abstract

Considering a Markov chain defined on a cycle, near-quadratic improvement of mixing is shown when only a subtle perturbation is introduced to the structure and non-reversible transition probabilities are used. More precisely, a mixing time of $O(n^{\frac{k+2}{k+1}})$ can be achieved by adding $k$ random edges to the cycle, keeping $k$ fixed while $n\to\infty$. The construction builds upon a biased random walk along the cycle.

Mixing on the cycle with constant size perturbation

Abstract

Considering a Markov chain defined on a cycle, near-quadratic improvement of mixing is shown when only a subtle perturbation is introduced to the structure and non-reversible transition probabilities are used. More precisely, a mixing time of can be achieved by adding random edges to the cycle, keeping fixed while . The construction builds upon a biased random walk along the cycle.

Paper Structure

This paper contains 6 sections, 14 theorems, 70 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Structure of typical tracks
  3. Spread of typical tracks
  4. Diffusive effect
  5. Numerical results
  6. Conclusions

Key Result

Theorem 1

Considering the upper bound: for any $\delta>0$ target exception probability there exist $\gamma(\delta)<\infty$ with the following property. For $n$ large enough and any $0<\varepsilon<1/2$, with probability at least $1-\delta$ in terms of the $k$ random extra edges we have Considering the lower bound: there exist $\gamma^*,\varepsilon^*>0$ with the following property. For $n$ large enough asymp

Figures (10)

  • Figure 1: Illustration of variables as composed in our model.
  • Figure 2: The shaded areas are the possible values that $y_1,y_2,y_3$ can take for $Y_m^{(-1,1,-1)}$
  • Figure 3: Intuitive illustration of $Y^{(1,-1,1)}_m$
  • Figure 4: Intuitive illustration of $Y_m(j)$
  • Figure 5: The red lines represents the location of $y'$ and $y"$. $y"-y'$ has the same "sign" indicator as $b$
  • ...and 5 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Remark 2
  • Definition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 19 more