Abstract: The frog model describes the spread of infection on a graph. Vertices initially contain independent Poisson numbers of particles with a common mean, the particle density. Only particles at the origin are active. Active particles perform independent random walks, awakening sleeping particles along the way. We study how quickly the infection spreads and, in the finite-lifespan model, the existence and sharpness of extinction–survival phase transitions in lifespan and particle density.

I will discuss activation times for the long-range frog model on $\mathbb Z^d$, explaining how a renormalization construction and a multiscale argument from long-range percolation yield activation times that grow polylogarithmically with distance from the origin when the jump-tail exponent lies strictly between 0 and the spatial dimension $d$. On vertex-transitive graphs of superlinear polynomial growth, the same renormalization, combined with a structural theorem for such graphs, establishes a nontrivial phase transition in particle lifespan at every fixed positive particle density. Finally, I will discuss sharpness of the phase transition on vertex-transitive graphs, including exponential tails for the number of visited vertices throughout the subcritical regime.

The talk is based on joint works with Omer Angel, Daniel de la Riva, and Jonathan Hermon.

Bio: Yuliang Shi is a PhD student in mathematics at the University of British Columbia, supervised by Omer Angel and Jonathan Hermon.