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Asymptotics of the Distribution of Distances between Random Points in a High-Dimensional Hypercube.

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Title: Asymptotics of the Distribution of Distances between Random Points in a High-Dimensional Hypercube.
Authors: Kislitsyn, A. A.1 (AUTHOR) alexey.kislitsyn@gmail.com, Orlov, Yu. N.1 (AUTHOR) ov3159f@yandex.ru, Tamm, D. T.1,2 (AUTHOR) tamm.dt@phystech.edu
Source: Computational Mathematics & Mathematical Physics. May2026, Vol. 66 Issue 5, p958-965. 8p.
Subjects: Hypercubes, Distribution (Probability theory), Geometry, Probability theory, Asymptotic analysis, Statistical sampling
Abstract: Statistical properties of distances between random points in an n-dimensional hypercube at large dimensions are investigated. An exact distance distribution function is constructed between random uniformly distributed points in the range of values r < 1 for an arbitrary dimension of the hypercube. An exact asymptotic distribution of distances between random points whose coordinates are independent and identically distributed is also constructed with increasing hypercube dimension. These results have two aspects of practical applications. First, in numerically processing large-dimensional random vectors, the task of constructing the distribution of distances between them is computationally difficult, so the theoretical function can serve as a good approximation and can significantly reduce the calculations. Second, in analyzing a large amount of data, the theoretical probability that the distance between them is less than one can be used as an estimate of their independence. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Statistical properties of distances between random points in an n-dimensional hypercube at large dimensions are investigated. An exact distance distribution function is constructed between random uniformly distributed points in the range of values r &lt; 1 for an arbitrary dimension of the hypercube. An exact asymptotic distribution of distances between random points whose coordinates are independent and identically distributed is also constructed with increasing hypercube dimension. These results have two aspects of practical applications. First, in numerically processing large-dimensional random vectors, the task of constructing the distribution of distances between them is computationally difficult, so the theoretical function can serve as a good approximation and can significantly reduce the calculations. Second, in analyzing a large amount of data, the theoretical probability that the distance between them is less than one can be used as an estimate of their independence. [ABSTRACT FROM AUTHOR]
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  Data: &lt;i&gt;Copyright of Computational Mathematics &amp; Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder&#39;s express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.&lt;/i&gt; (Copyright applies to all Abstracts.)
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      – SubjectFull: Asymptotic analysis
        Type: general
      – SubjectFull: Statistical sampling
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      – TitleFull: Asymptotics of the Distribution of Distances between Random Points in a High-Dimensional Hypercube.
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              Text: May2026
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