Abstract
Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi’s parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or “car”) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet.
We provide an explicit expression for the so-called parking constant in terms of the generating function. That is, the occupation probability, averaged over dynamics and the probability distribution of the random trees converges in the large-time limit to (1−α 2)/2 with \(\int_{\alpha}^{1}\frac{xdx}{G(x)}=1\) .
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Dehling, H.G., Fleurke, S.R. & Külske, C. Parking on a Random Tree. J Stat Phys 133, 151–157 (2008). https://doi.org/10.1007/s10955-008-9589-9
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DOI: https://doi.org/10.1007/s10955-008-9589-9