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Pinsky, Ross G.

The infinite limit of separable permutations

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  • Media type: E-Article
  • Title: The infinite limit of separable permutations
  • Contributor: Pinsky, Ross G.
  • imprint: Wiley, 2021
  • Published in: Random Structures & Algorithms
  • Language: English
  • DOI: 10.1002/rsa.21014
  • ISSN: 1042-9832; 1098-2418
  • Origination:
  • Footnote:
  • Description: <jats:title>Abstract</jats:title><jats:p>Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0001.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0001" /> denote the uniform probability measure on the set of separable permutations in <jats:italic>S</jats:italic><jats:sub><jats:italic>n</jats:italic></jats:sub>. Let <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0002.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0002" /> with an appropriate metric and denote by <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0003.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0003" /> the compact metric space consisting of functions <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0004.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0004" /> from <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0005.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0005" /> to <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0006.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0006" /> which are injections when restricted to <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0007.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0007" />; that is, if <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0008.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0008" />, <jats:italic>i</jats:italic> ≠ <jats:italic>j</jats:italic>, then <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0009.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0009" />. Extending permutations <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0010.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0010" /> by defining <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0011.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0011" />, for <jats:italic>j</jats:italic> &gt; <jats:italic>n</jats:italic>, we have <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0012.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0012" />. We show that <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0013.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0013" /> converges weakly on <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/rsa21014-math-0014.png" xlink:title="urn:x-wiley:rsa:media:rsa21014:rsa21014-math-0014" /> to a limiting distribution of regenerative type, which we calculate explicitly.</jats:p>