Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Thursday, September 25, 2025, 12:15 pm

Duration: 30 minutes

Location: OAT S15

Speaker: Kosta Panagiotou (LMU)

Dispersion on the complete graph

We study a synchronous process of particles moving on the vertices of a graph G. Initially, M particles are placed on a vertex. 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 talk we consider the case where G = K_n, a complete graph on n vertices, and M = cn. In that case, the dispresion time undergoes a phase transition from logarithmic to exponential time when c is around 1/2. We will investigate the fine details of this transition, and we will establish that there is a critical window of order n^{1/2}, where the dispersion time is also of that order. Moreover, we will derive very explicit descriptions of the distributions of various quantities, like the dispersion time and the total number of jumps. Within the proof we develop an explicit strategy that is based on diffusion approximation, a powerful tool from stochastic analysis, that allow us to describe the behavior of processes that do not concentrate around a deterministic trajectory, and that is very likely to be of great use in other problems. This is joint work with U. De Ambroggio, T. Makai and A. Steibel.

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