Parking on supercritical Galton-Watson tree

Abstract

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 arrive at. Let X be the total number of cars that arrive at 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 and P (X = 0) are discontinuous at the critical threshold, describe the growth rate of EX above criticality, and prove that X stochastically increases as the initial car arrival distribution becomes less concentrated. We also provide a new characterization of the threshold with a generating function condition satisfied by the time of first arrival at the root. For the simple 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 also prove that the location of the phase transition depends on more than just the mean number of cars arriving to each vertex.

Diffusion-limited annihilating systems and the increasing convex order

“…They were interested in these dynamics because of a connection to a spin glass model. For the usual parking process Bahl, Barnet, and Junge demonstrated through an example on d-ary trees that the phase state depends on more than just the mean initial density of particles [BBJ21,Proposition 7]. In an attempt to qualitatively describe this phenomenon [BBJ21, Theorem 8] further showed that the total number of visits to the root increases when the underlying placement of A-particles is made more volatile.…”