Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

Nayantara Bhatnagar; Nick Crawford; Elchanan Mossel; Arnab Sen

doi:10.1007/s11083-011-9244-y

Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

Nayantara Bhatnagar, Nick Crawford, Elchanan Mossel, Arnab Sen

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0, 1], which follows the Rayleigh distribution.

Original language	English (US)
Pages (from-to)	289-311
Number of pages	23
Journal	Order
Volume	30
Issue number	1
DOIs	https://doi.org/10.1007/s11083-011-9244-y
State	Published - Mar 2013

Bibliographical note

Funding Information:
A. Sen was supported by DOD ONR grant N0014-07-1-05-06, DMS 0528488, and DMS 0548249 (CAREER).

Funding Information:
N. Bhatnagar was supported by DOD ONR grant N0014-07-1-05-06 and DMS 0528488.

Funding Information:
N. Crawford was supported by DMS 0548249 (CAREER).

Keywords

Brownian excursion
Random posets
Scaling limits
Width-2 partial order

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10.1007/s11083-011-9244-y

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title = "Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window",

abstract = "We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0, 1], which follows the Rayleigh distribution.",

keywords = "Brownian excursion, Random posets, Scaling limits, Width-2 partial order",

author = "Nayantara Bhatnagar and Nick Crawford and Elchanan Mossel and Arnab Sen",

note = "Funding Information: A. Sen was supported by DOD ONR grant N0014-07-1-05-06, DMS 0528488, and DMS 0548249 (CAREER). Funding Information: N. Bhatnagar was supported by DOD ONR grant N0014-07-1-05-06 and DMS 0528488. Funding Information: N. Crawford was supported by DMS 0548249 (CAREER).",

year = "2013",

month = mar,

doi = "10.1007/s11083-011-9244-y",

language = "English (US)",

volume = "30",

pages = "289--311",

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T1 - Scaling Limits for Width Two Partially Ordered Sets

T2 - The Incomparability Window

AU - Bhatnagar, Nayantara

AU - Crawford, Nick

AU - Mossel, Elchanan

AU - Sen, Arnab

N1 - Funding Information: A. Sen was supported by DOD ONR grant N0014-07-1-05-06, DMS 0528488, and DMS 0548249 (CAREER). Funding Information: N. Bhatnagar was supported by DOD ONR grant N0014-07-1-05-06 and DMS 0528488. Funding Information: N. Crawford was supported by DMS 0548249 (CAREER).

PY - 2013/3

Y1 - 2013/3

N2 - We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0, 1], which follows the Rayleigh distribution.

AB - We study the structure of a uniformly randomly chosen partial order of width 2 on n elements. We show that under the appropriate scaling, the number of incomparable elements converges to the height of a one dimensional Brownian excursion at a uniformly chosen random time in the interval [0, 1], which follows the Rayleigh distribution.

KW - Brownian excursion

KW - Random posets

KW - Scaling limits

KW - Width-2 partial order

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DO - 10.1007/s11083-011-9244-y

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VL - 30

SP - 289

EP - 311

JO - Order

JF - Order

IS - 1

ER -

Scaling Limits for Width Two Partially Ordered Sets: The Incomparability Window

Abstract

Bibliographical note

Keywords

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