Unified derivation of the limit shape for multiplicative ensembles of random integer partitions with equiweighted parts12. Issue 2 (18th April 2014)

• Record Type: Journal Article
• Authors: Bogachev, Leonid V.
• Is Part Of: Random structures & algorithms. Volume 47:Issue 2(2015)
• Page Start: 227
• Page End: 266
• Publication Date: 2014-04-18
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20540
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We derive the limit shape of Young diagrams, associated with growing integer partitions, with respect to multiplicative probability measures underpinned by the generating functions of the form <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twghz0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ℱ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∏</mml:mo><mml:mrow><mml:mo>ℓ</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>∞</mml:mo></mml:munderover><mml:mrow><mml:msub><mml:mo>ℱ</mml:mo><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>z</mml:mi><mml:mo>ℓ</mml:mo></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> (which entails equal weighting among possible parts <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj02" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ℓ</mml:mo><mml:mo>∈</mml:mo><mml:mo>ℕ</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>). Under mild technical assumptions on the function <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj1m" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>ln</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mo>ℱ</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, we show that the limit shape <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgj25" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ω</mml:mo><mml:mo>*</mml:mo></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> exists and is given by the equation <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twghb2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mo>γ</mml:mo><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi mathvariant="normal">e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mo>γ</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj23twgh8z" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20540:rsa20540-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>γ</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:mrow><mml:msubsup><mml:mo>∫</mml:mo><mml:mn>0</mml:mn><mml:mn>1</mml:mn></mml:msubsup><mml:mrow><mml:msup><mml:mi>u</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:mstyle><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo> </mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi>u</mml:mi></mml:mrow></mml:math></alternatives></inline-formula>. The wide class of partition measures covered by this result includes (but is not limited to) representatives of the three meta‐types of decomposable combinatorial structures — assemblies, multisets, and selections. Our method is based on the usual randomization and conditioning; to this end, a suitable local limit theorem is proved. The proofs are greatly facilitated by working with the cumulants of sums of the part counts rather than with their moments.Copyright © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 227–266, 2015</p> </abstract>