Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dispersion on the Complete Graph

Umberto De Ambroggio, Tamás Makai, Konstantinos Panagiotou

TL;DR

The paper analyzes the dispersion time $T_{n,M}$ for the synchronous dispersion process on the complete graph in the critical regime $M=(1+\varepsilon)n/2$ with $|\varepsilon|=o(1)$. It develops a drift-based framework showing $\mathbb{E}[U_{t+1}-U_t|U_t]\;=\;\varepsilon U_t-\Theta(U_t^2/n)$ and couples the unhappy-particle dynamics to a binomial/branching process to control upper and lower tails. The main results establish three distinct behaviors: near criticality the dispersion time is $\Theta(n^{1/2})$ within a window of size $O(\sqrt{n})$, while moving away from criticality yields transitions to logarithmic or exponential scaling, with precise tail bounds presented. The methods combine drift analysis, concentration inequalities, and branching-process couplings, providing sharp probabilistic bounds and highlighting a smooth phase transition in dispersion time as $\varepsilon$ varies. The findings offer a detailed quantitative picture of the critical window and set the stage for distributional limit results in the scaling regime.

Abstract

We consider a synchronous process of particles moving on the vertices of a graph $G$, introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, $M$ particles are placed on a vertex of $G$. At the beginning of each time step, for every vertex inhabited by at least two particles, each of these particles moves independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle. Cooper et al. showed that when the underlying graph is the complete graph on~$n$ vertices, then there is a phase transition when the number of particles $M = n/2$. They showed that if $M<(1-\varepsilon)n/2$ for some fixed $\varepsilon>0$, then the process finishes in a logarithmic number of steps, while if $M>(1+\varepsilon)n/2$, an exponential number of steps are required with high probability. Here we provide a thorough asymptotic analysis of the dispersion time around criticality, where $\varepsilon = o(1)$, and describe the transition from logarithmic to exponential time. As a consequence of our results we establish, for example, that the dispersion time is in probability and in expectation in $Θ(n^{1/2})$ when $|\varepsilon| = O(n^{-1/2})$, and provide qualitative bounds for its tail behavior.

Dispersion on the Complete Graph

TL;DR

The paper analyzes the dispersion time for the synchronous dispersion process on the complete graph in the critical regime with . It develops a drift-based framework showing and couples the unhappy-particle dynamics to a binomial/branching process to control upper and lower tails. The main results establish three distinct behaviors: near criticality the dispersion time is within a window of size , while moving away from criticality yields transitions to logarithmic or exponential scaling, with precise tail bounds presented. The methods combine drift analysis, concentration inequalities, and branching-process couplings, providing sharp probabilistic bounds and highlighting a smooth phase transition in dispersion time as varies. The findings offer a detailed quantitative picture of the critical window and set the stage for distributional limit results in the scaling regime.

Abstract

We consider a synchronous process of particles moving on the vertices of a graph , introduced by Cooper, McDowell, Radzik, Rivera and Shiraga (2018). Initially, particles are placed on a vertex of . At the beginning of each time step, for every vertex inhabited by at least two particles, each of these particles moves independently to a neighbour chosen uniformly at random. The process ends at the first step when no vertex is inhabited by more than one particle. Cooper et al. showed that when the underlying graph is the complete graph on~ vertices, then there is a phase transition when the number of particles . They showed that if for some fixed , then the process finishes in a logarithmic number of steps, while if , an exponential number of steps are required with high probability. Here we provide a thorough asymptotic analysis of the dispersion time around criticality, where , and describe the transition from logarithmic to exponential time. As a consequence of our results we establish, for example, that the dispersion time is in probability and in expectation in when , and provide qualitative bounds for its tail behavior.
Paper Structure (15 sections, 22 theorems, 153 equations)

This paper contains 15 sections, 22 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Model & Results.
  3. Directions for Future Research.
  4. Related Work.
  5. Main Proof Ideas
  6. Notation.
  7. Preliminaries
  8. Drift Analysis, Martingale Estimates & Bernoulli's inequalities
  9. Bounds on $\mathbb{E}[U_{t+1} \mid U_t]$
  10. Reducing the Number of Unhappy Particles
  11. Branching Process Approximation
  12. The Upper Tail
  13. Proof of Proposition \ref{['mainpropsec3']}.
  14. The Lower Tail
  15. Proof of Proposition \ref{['mainpropsec4']}.

Key Result

Theorem 1.1

There is a $C > 0$ such that the following is true for sufficiently large $n$ and all $A \ge 1$. Let $M = (1+\varepsilon)n/2 \in \mathbb{N}$, where $\varepsilon = o(1)$ and $|\varepsilon| < 1/9$. If $\varepsilon< -e n^{-1/2}$, then Moreover, if $|\varepsilon| \le e n^{-1/2}$, then Finally, if $\varepsilon > e n^{-1/2}$, then

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Thm. 10 in KK19
  • Theorem 2.2: Azuma-Hoeffding's inequality, Thm. 23.16 in FKIRG
  • Theorem 2.3: Bernoulli's inequalities, (5.3.78-79) in MR3408971
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 27 more