Tobias Johnson

From transience to recurrence with Poisson tree frogs

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.

Recurrence and transience for the frog model on trees

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 dd-ary tree. We prove the model undergoes a phase transition, finding it recurrent for d=2d=2 and transient for d≥5d≥5. Simulations suggest strong recurrence for d=2d=2, weak recurrence for d=3d=3, and transience for d≥4d≥4. Additionally, we prove a 0–1 law for all dd-ary trees, and we exhibit a graph on which a 0–1 law does not hold. To prove recurrence when d=2d=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=5d=5 relies on computer calculations for the transition probabilities of a large Markov chain. We also include the proof for d≥6d≥6, which uses similar techniques but does not require computer assistance.

Functional limit theorems for random regular graphs

Consider

Cycles and eigenvalues of sequentially growing random regular graphs

d

The critical density for the frog model is the degree of the tree

uniformly random permutation matrices on

Bounds to the normal for proximity region graphs

n