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- W4287978647 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 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." @default.
- W4287978647 created "2022-07-26" @default.
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- W4287978647 date "2019-12-30" @default.
- W4287978647 modified "2023-09-28" @default.
- W4287978647 title "Parking on supercritical Galton-Watson trees" @default.
- W4287978647 doi "https://doi.org/10.48550/arxiv.1912.13062" @default.
- W4287978647 hasPublicationYear "2019" @default.
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