Probabilizing parking functions

Diaconis, Persi; Hicks, Angela

doi:10.1016/j.aam.2017.05.004

Cited by 31 publications

(61 citation statements)

References 45 publications

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“…Their relation to polytopes was studied by Stanley and Pitman in [SP02]. More recently, Diaconis and Hicks studied the geometry of a random parking function [DH17].…”

Section: Introductionmentioning

confidence: 99%

Parking on transitive unimodular graphs

Damron¹,

Gravner²,

Junge³

et al. 2019

Ann. Appl. Probab.

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Place a car independently with probability p at each site of a graph. Each initially vacant site is a parking spot that can fit one car. Cars simultaneously perform independent random walks. When a car encounters an available parking spot it parks there. Other cars can still drive over the site, but cannot park there. For a large class of transitive and unimodular graphs, we show that the root is almost surely visited infinitely many times when p ≥ 1/2, and only finitely many times otherwise.2010 Mathematics Subject Classification. 60K35, 82C22,82B26.

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“…Their relation to polytopes was studied by Stanley and Pitman in [SP02]. More recently, Diaconis and Hicks studied the geometry of a random parking function [DH17].…”

Section: Introductionmentioning

confidence: 99%

Parking on transitive unimodular graphs

Damron¹,

Gravner²,

Junge³

et al. 2019

Ann. Appl. Probab.

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“…, n − k + 1 that can be constructed by the following procedure. Steps (1) and (4) involve no choice, while there are n−m−k 2 ,..., m+1 possibilities for step (2) and m+1…”

Section: Parking Completions Of a Block (Proof Of Corollary 12)mentioning

confidence: 99%

“…. , ), and Diaconis-Hicks [4], who considered the case that t = (i) for some i ∈ [n]. Although it is not immediate, Theorem 1.1 implies the following generalization of Diaconis and Hicks' enumeration of parking function shuffles [ For increasing parking completions, there is a similar result.…”

Section: Introductionmentioning

confidence: 97%

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Enumerating Parking Completions Using Join and Split

Adeniran¹,

Butler²,

Dorpalen-Barry³

et al. 2020

Electron. J. Combin.

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Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\leqslant i\}|+|\{c\in \mathbf{c}\mid c\leqslant i\}|\geqslant i$ for all $i$ in $[n]$. We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any. A parking completion corresponds to a set of preferences $\mathbf{c}$ where all cars park. We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split. Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the \emph{signature parking functions} of Ceballos and González D'León.

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“…There has been significant followup study of the combinatorial structures that arise from parking functions. See the work of Stanley and Pitman (Stanley, 1997(Stanley, , 1998Stanley and Pitman, 2002) as well as Diaconis and Hicks (2017). There are also connections to the Abelian sandpile and activated random walk (Chebikin and Pylyavskyy, 2005;Cabezas et al, 2014a).…”

Section: Introductionmentioning

confidence: 99%

Parking on supercritical Galton-Watson tree

Bahl¹,

Barnet²,

Junge³

2021

ALEA

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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.

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Probabilizing parking functions

Cited by 31 publications

References 45 publications

Parking on transitive unimodular graphs

Parking on transitive unimodular graphs

Enumerating Parking Completions Using Join and Split

Parking on supercritical Galton-Watson tree

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