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$L^2$-cutoff for the averaging process on random regular graphs

Pietro Caputo, Matteo Quattropani, Federico Sau

Abstract

We study the mixing time of the averaging process on a large random $d$-regular graph, $d\ge 3$, and prove an $L^2$-cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree $d=10$: for small degrees, i.e., $d\le 10$, the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for $d> 10$ its $L^2$-mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the $L^1$-mixing.

$L^2$-cutoff for the averaging process on random regular graphs

Abstract

We study the mixing time of the averaging process on a large random -regular graph, , and prove an -cutoff with an explicit cutoff time. Somewhat surprisingly, we uncover a phase transition at the finite, fixed degree : for small degrees, i.e., , the averaging process mixes as fast as the corresponding random walk on the same graph, whereas for its -mixing is governed by a different, slower mechanism. Our proof relies on a detailed asymptotic analysis of an auxiliary biased birth-and-death chain with a slow bond. We also briefly discuss an analogous phase transition for the -mixing.
Paper Structure (14 sections, 7 theorems, 77 equations, 2 figures)

This paper contains 14 sections, 7 theorems, 77 equations, 2 figures.

Table of Contents

  1. Introduction, model, and main result
  2. Main result
  3. Further remarks and open problems
  4. Comparison with the random walk
  5. Large-degree regime
  6. $L^p$-cutoff and phase transitions
  7. Proof of Theorem \ref{['th:L2']}
  8. Coupled random walks and the slow-bond birth-and-death chain
  9. Lower bound
  10. Upper bound
  11. Proof of Proposition \ref{['prop:CRW-infinite-tree-CT']}
  12. Preliminaries
  13. Proof of Proposition \ref{['prop:CRW-infinite-tree']}
  14. From discrete- to continuous-time: proof of Proposition \ref{['prop:CRW-infinite-tree-CT']}

Key Result

Theorem 1.1

Let, for every integer $d\ge 3$, where Further, define (note that $\gamma \in (0,1)$, for all $d\ge 3$; see eq:gamma-properties below) Then, for every $\varepsilon> 0$, we have

Figures (2)

  • Figure 1.1: Left: Plot of the function $d\mapsto \gamma_d$, for $d\ge 3$. Right: Plots of the functions $d\mapsto \varrho_d$ (blue) and $d\mapsto \sigma_d$ (orange), for $d\ge 10$.
  • Figure 3.1: The horizontal axis corresponds to the variable $d\ge 3$. The function in red is constantly equal to $1$. The orange curve is the function $d\mapsto |z_2|$, while the blue one is $d\mapsto |z_1|$ (note that the two curves coincide for $d\le 9$). In green, the curve $\varrho^{-1}=d/(2\sqrt{d-1})$.

Theorems & Definitions (19)

  • Theorem 1.1: $L^2$-cutoff
  • Definition 2.1: Slow-bond BD chain
  • Proposition 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof : Proof of Theorem \ref{['th:L2']} (lower bound)
  • Lemma 2.5
  • proof : Proof of Proposition \ref{['prop:lb']}
  • proof : Proof of Lemma \ref{['lem:lb']}
  • proof : Proof of Theorem \ref{['th:L2']} (upper bound)
  • ...and 9 more