Consider the following interacting particle system on the $d$-ary tree, known as the frog model: Initially, one particle is awake at the root and i.i.d. Poisson many particles are sleeping at every other vertex. Particles that are awake perform simple random walks, awakening any sleeping particles they encounter. We prove that there is a phase transition between transience and recurrence as the initial density of particles increases, and we give the order of the transition up to a logarithmic factor.Comment: Published at http://dx.doi.org/10.1214/15-AAP1127 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The frog model is a growing system of random walks where a particle is added whenever a new site is visited. A longstanding open question is how often the root is visited on the infinite d-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d = 2 and transient for d ≥ 5. Simulations suggest strong recurrence for d = 2, weak recurrence for d = 3, and transience for d ≥ 4. Additionally, we prove a 0-1 law for all d-ary trees, and we exhibit a graph on which a 0-1 law does not hold.To prove recurrence when d = 2, we construct a recursive distributional equation for the number of visits to the root in a smaller process and show the unique solution must be infinity a.s. The proof of transience when d = 5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d ≥ 6, which uses similar techniques but does not require computer assistance.

We develop an agent-based model on a network meant to capture features unique to COVID-19 spread through a small residential college. We find that a safe reopening requires strong policy from administrators combined with cautious behavior from students. Strong policy includes weekly screening tests with quick turnaround and halving the campus population. Cautious behavior from students means wearing facemasks, socializing less, and showing up for COVID-19 testing. We also find that comprehensive testing and facemasks are the most effective single interventions, building closures can lead to infection spikes in other areas depending on student behavior, and faster return of test results significantly reduces total infections.

At each site of a supercritical Galton-Watson tree place a parking spot which can accommodate one car. Initially, an independent and identically distributed number of cars arrive at each vertex. Cars proceed towards the root in discrete time and park in the first available spot they come to. Let X be the total number of cars that arrive to the root. Goldschmidt and Przykucki proved that X undergoes a phase transition from being finite to infinite almost surely as the mean number of cars arriving to each vertex increases. We show that EX is finite at the critical threshold, describe its growth rate above criticality, and prove that it increases as the initial car arrival distribution becomes less concentrated. For the canonical case that either 0 or 2 cars arrive at each vertex of a d-ary tree, we give improved bounds on the critical threshold and show that P (X = 0) is discontinuous as a function of α at αc.

We consider evoSIR, a variant of the SIR model, on Erdős-Renyi random graphs in which susceptibles with an infected neighbor break that connection at rate ρ and rewire to a randomly chosen individual. We compute the critical infection rate λ c and the probability of a large epidemic by showing that they are the same for the delSIR model in which S − I connections are deleted instead of rewired. The final size of a large delSIR epidemic has a continuous transition. Simulations suggest that the final size of a large evoSIR epidemic is discontinuous at λ c .