Birthday inequalities, repulsion, and hard spheres

Perkins, Will

doi:10.1090/proc/13028

Cited by 9 publications

(8 citation statements)

References 20 publications

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“…Despite extensive study, the exact solution is known only for dimensions d = 1, 2, 3, 8 and 24. See [8,16] for rigorous bounds on packing densities in general dimension d.…”

Section: Introductionmentioning

confidence: 99%

Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

Guo,

Jerrum

2018

Preprint

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We present a perfect simulation of the hard disks model via the partial rejection sampling method. Provided the density of disks is not too high, the method produces exact samples in O(log n) rounds, and total time O(n), where n is the expected number of disks. The method extends easily to the hard spheres model in d > 2 dimensions. In order to apply the partial rejection method to this continuous setting, we provide an alternative perspective of its correctness and run-time analysis that is valid for general state spaces.

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“…Despite extensive study, the exact solution is known only for dimensions d = 1, 2, 3, 8 and 24. See [8,16] for rigorous bounds on packing densities in general dimension d.…”

Section: Introductionmentioning

confidence: 99%

Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

Guo,

Jerrum

2018

Preprint

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“…Here, it holds , and it can be shown without much effort that is indeed an upper bound and a good estimate for the probability that are pairwise different (see, e.g. [3,4]). For example, with and , it holds…”

Section: Expected Number Of Packs Needed For a Matchmentioning

confidence: 84%

A probabilistic way to discover the rainbow

Prochno¹,

Schmitz²

2022

Math. Gaz.

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A slogan that you find on the back of a pack of Skittles candy says ‘No two rainbows are the same. Neither are two packs of Skittles. Enjoy an odd mix.’ An online blog [1] describes how the blogger found two identical packs of Skittles, among 468 packs with a total of 27,740 Skittles. Meticulously collecting the data for this experiment was apparently triggered by some earlier calculations. More precisely, the blogger writes:

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“…Conjecture also appeared in a draft of . These conjectures are stronger than Theorems and and imply the conjectures of and .…”

Section: Discussionmentioning

confidence: 99%

Independent sets, matchings, and occupancy fractions

Davies

Jenssen

Perkins

et al. 2017

Journal of London Math Soc

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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d‐regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of Kd,d maximizes the number of independent sets and the independence polynomial of a d‐regular graph. For matchings, this shows that the matching polynomial and the total number of matchings of a d‐regular graph are maximized by a union of copies of Kd,d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström. In probabilistic language, our main theorems state that for all d‐regular graphs and all λ, the occupancy fraction of the hard‐core model and the edge occupancy fraction of the monomer‐dimer model with fugacity λ are maximized by Kd,d. Our method involves constrained optimization problems over distributions of random variables and applies to all d‐regular graphs directly, without a reduction to the bipartite case.

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Birthday inequalities, repulsion, and hard spheres

Cited by 9 publications

References 20 publications

Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

Perfect simulation of the Hard Disks Model by Partial Rejection Sampling

A probabilistic way to discover the rainbow

Independent sets, matchings, and occupancy fractions

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