Limit Laws for Critical Dispersion on Complete Graphs

Authors Umberto De Ambroggio, Tamás Makai, Konstantinos Panagiotou, Annika Steibel



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Author Details

Umberto De Ambroggio
  • Department of Mathematics, LMU Munich, Germany
Tamás Makai
  • Department of Mathematics, LMU Munich, Germany
Konstantinos Panagiotou
  • Department of Mathematics, LMU Munich, Germany
Annika Steibel
  • Department of Mathematics, LMU Munich, Germany

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Umberto De Ambroggio, Tamás Makai, Konstantinos Panagiotou, and Annika Steibel. Limit Laws for Critical Dispersion on Complete Graphs. In 35th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 302, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.AofA.2024.26

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. In subsequent time steps, all particles that are located on a vertex inhabited by at least two particles jump 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; we call this (random) time step the dispersion time.
In this work we study the case where G is the complete graph on n vertices and the number of particles is M = n/2+α n^{1/2} + o(n^{1/2}), α ∈ ℝ. This choice of M corresponds to the critical window of the process, with respect to the dispersion time. We show that the dispersion time, if rescaled by n^{-1/2}, converges in p-th mean, as n → ∞ and for any p ∈ ℝ, to a continuous and almost surely positive random variable T_α. We find that T_α is the absorption time of a standard logistic branching process, thoroughly investigated by Lambert (2005), and we determine its expectation. In particular, in the middle of the critical window we show that 𝔼[T₀] = π^{3/2}/√7, and furthermore we formulate explicit asymptotics when |α| gets large that quantify the transition into and out of the critical window. We also study the random variable counting the total number of jumps that are performed by the particles until the dispersion time is reached and prove that, if rescaled by nln(n), it converges to 2/7 in probability.

Subject Classification

ACM Subject Classification
  • Theory of computation → Random walks and Markov chains
Keywords
  • Random processes on graphs
  • diffusion processes
  • stochastic differential equations
  • martingale inequalities

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References

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