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Random Partial Orders Defined by Angular Domains

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Abstract

The d-dimensional random partial order is the intersection of d independently and uniformly chosen (with replacement) linear orders on the set [n] = {1, 2, . . . , n}. This is equivalent to picking n points uniformly at random in the d-dimensional unit cube \(Q_d=[0,1]^d\) with the coordinate-wise ordering. If d = 2, then this can be rephrased by declaring that for any pair P 1, P 2 ∈ Q 2 we have P 1 ≺ P 2 if and only if P 2 lies in the positive upper quadrant defined by the two axis-parallel lines crossing at P 1. In this paper we study the random partial order with parameter α (0 ≤ α ≤ π) which is generated by picking n points uniformly at random from Q 2 equipped with the same partial order as above but with the quadrant replaced by an angular domain of angle α.

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Correspondence to Balázs Patkós.

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P. Balister was partially supported by NSF Grant CCF-0728928.

B. Patkós was supported by NSF Grant CCF-0728928.

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Balister, P., Patkós, B. Random Partial Orders Defined by Angular Domains. Order 28, 341–355 (2011). https://doi.org/10.1007/s11083-010-9172-2

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  • DOI: https://doi.org/10.1007/s11083-010-9172-2

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