Parking on transitive unimodular graphs

Damron, Michael; Gravner, Janko; Junge, Matthew; Lyu, Hanbaek; Sivakoff, David

doi:10.1214/18-aap1443

Cited by 11 publications

(21 citation statements)

References 29 publications

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“…Remark For p < p 0 we obtain

𝔼_{p} τ < \infty

and consequently, by a mass transport argument (Lemma 3.1)

𝔼_{p} V < \infty

, where V is the total number of visits to the origin by cars. In [5], this was shown for p < (256 d 6 e 2 ) −1 ≍ d −6 . Our value of p 0 ≍ d −2 improves on this bound.…”

Section: Statement Of Resultsmentioning

confidence: 84%

“…In this note, we study the tail of the distribution of the parking time of cars in the subcritical phase for the parking model on

ℤ^{d}

. The model was introduced in [5] and studied further in [14]; a similar continuous‐time particle system was introduced and studied earlier in [3, 4]. It is roughly defined as follows.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…If a car visits an occupied spot, it does not park, and moves at the next timestep. We refer the reader to [5] for a more precise definition of the process. Our probability space is defined in Section 2.1.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…In [3–5] it was shown that there is a phase transition in p : a.s. the origin is visited by finitely many distinct cars for p < 1/2 (subcritical phase) and infinitely many for p ≥ 1/2 (critical and supercritical phases). Furthermore, the parking time for the car of the origin, if it exists, is finite a.s. for p ≤ 1/2, and infinite with positive probability when p > 1/2.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Stretched exponential decay for subcritical parking times on

Damron

Lyu

Sivakoff

2021

Random Struct Algorithms

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At each vertex of ℤd, place a car with probability p or vacant parking spot with probability 1 − p. Cars perform independent random walks and park at vacant spots, rendering them passable. There is a transition at p = 1/2: the origin is a.s. visited by finitely many distinct cars when p < 1/2, and by infinitely many when p ≥ 1/2. Furthermore, a.s. all cars park if p ≤ 1/2 and some never park for p > 1/2. For small p, we prove that the parking time τ of the car initially at the origin satisfies exp−C1tdd+2≤ℙp(τ>t)≤exp−c2tdd+2. For d = 1, these inequalities hold for p < 1/2. In contrast, when p > 1/2 and d = 1, methods of Bramson–Lebowitz imply that the tail of the parking time of the spot of the origin decays like e−ct. Our exponent d/(d + 2) also differs from those previously obtained in the case of moving obstacles.

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“…Remark For p < p 0 we obtain

𝔼_{p} τ < \infty

and consequently, by a mass transport argument (Lemma 3.1)

𝔼_{p} V < \infty

, where V is the total number of visits to the origin by cars. In [5], this was shown for p < (256 d 6 e 2 ) −1 ≍ d −6 . Our value of p 0 ≍ d −2 improves on this bound.…”

Section: Statement Of Resultsmentioning

confidence: 84%

“…In this note, we study the tail of the distribution of the parking time of cars in the subcritical phase for the parking model on

ℤ^{d}

. The model was introduced in [5] and studied further in [14]; a similar continuous‐time particle system was introduced and studied earlier in [3, 4]. It is roughly defined as follows.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Section: Statement Of Resultsmentioning

confidence: 99%

Section: Statement Of Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Stretched exponential decay for subcritical parking times on

Damron

Lyu

Sivakoff

2021

Random Struct Algorithms

Self Cite

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show abstract

“…One model where interaction between particles prevents settling at a cite is a two-type particle system called "Oil and Water" where particles of opposing types displace each other [13]. There have been some papers on models related to the Parking function of a graph where cars drive randomly around a graph searching for vacant spots [16,21]. More commonly, however, interaction is directly between particles and not with the host graph such as predator prey/coalescing models [15].…”

Section: Related Workmentioning

confidence: 99%

The Dispersion Time of Random Walks on Finite Graphs

Rivera

Sauerwald

Stauffer

et al. 2019

The 31st ACM Symposium on Parallelism in Algorithms and Architectures

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We study two random processes on an n-vertex graph inspired by the internal diffusion limited aggregation (IDLA) model. In both processes n particles start from an arbitrary but fixed origin. Each particle performs a simple random walk until first encountering an unoccupied vertex, and at which point the vertex becomes occupied and the random walk terminates. In one of the processes, called Sequential-IDLA, only one particle moves until settling and only then does the next particle start whereas in the second process, called Parallel-IDLA, all unsettled particles move simultaneously. Our main goal is to analyze the so-called dispersion time of these processes, which is the maximum number of steps performed by any of the n particles.In order to compare the two processes, we develop a coupling that shows the dispersion time of the Parallel-IDLA stochastically dominates that of the Sequential-IDLA; however, the total number of steps performed by all particles has the same distribution in both processes. This coupling also gives us that dispersion time of Parallel-IDLA is bounded in expectation by dispersion time of the Sequential-IDLA up to a multiplicative log n factor. Moreover, we derive asymptotic upper and lower bound on the dispersion time for several graph classes, such as cliques, cycles, binary trees, d-dimensional grids, hypercubes and expanders. Most of our bounds are tight up to a multiplicative constant.

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Non-intersection of transient branching random walks

Hutchcroft

2020

Probab. Theory Relat. Fields

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Let G be a Cayley graph of a nonamenable group with spectral radius ρ < 1. It is known that branching random walk on G with offspring distribution µ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring µ satisfies µ ≤ ρ −1 . Benjamini and Müller (Groups Geom. Dyn., 2010) conjectured that throughout the transient supercritical phase 1 < µ ≤ ρ −1 , and in particular at the recurrence threshold µ = ρ −1 , the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors.

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Parking on transitive unimodular graphs

Cited by 11 publications

References 29 publications

Stretched exponential decay for subcritical parking times on

Stretched exponential decay for subcritical parking times on

The Dispersion Time of Random Walks on Finite Graphs

Non-intersection of transient branching random walks

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