Asymptotics of trees with a prescribed degree sequence and applications. Issue 3 (5th September 2012)

• Record Type: Journal Article
• Authors: Broutin, Nicolas; Marckert, Jean‐François
• Is Part Of: Random structures & algorithms. Volume 44:Issue 3(2014)
• Page Start: 290
• Page End: 316
• Publication Date: 2012-09-05
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20463
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Let <italic>t</italic> be a rooted tree and <italic>n</italic><sub><italic>bi</italic></sub>(<italic>t</italic>) the number of nodes in <italic>t</italic> having <italic>i</italic> children. The degree sequence <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10bn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>, </mml:mo><mml:mi>i</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> of <italic>t</italic> satisfies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10c6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:munder><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:mstyle><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mstyle displaystyle="true"><mml:munder><mml:mo>∑</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:munder><mml:mi>i</mml:mi></mml:mstyle><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo>|</mml:mo><mml:mi>t</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives>, where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10dr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>|</mml:mo><mml:mi>t</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives> denotes the number of nodes in <italic>t</italic>. In this paper, we consider trees sampled uniformly among all plane trees having the same degree sequence <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10f9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">s</mml:mi></mml:math></alternatives>; we write <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10vx" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>ℙ</mml:mo><mml:mi mathvariant="bold">s</mml:mi></mml:msub></mml:mrow></mml:math></alternatives> for the corresponding distribution. Let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10tc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="bold">s</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>κ</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mo>κ</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>, </mml:mo><mml:mi>i</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> be a list of degree sequences indexed by κ corresponding to trees with size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10x1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mo>κ</mml:mo></mml:msub><mml:mo>→</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math></alternatives>. We show that under some simple and natural hypotheses on <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10wg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="bold">s</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>κ</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>, </mml:mo><mml:mo>κ</mml:mo><mml:mo>></mml:mo><mml:mn>0</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> the trees sampled under <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s10zk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mo>ℙ</mml:mo><mml:mrow><mml:mi mathvariant="bold">s</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mo>κ</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msub></mml:mrow></mml:math></alternatives> converge to the Brownian continuum random tree after normalisation by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4x9s114v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20463:rsa20463-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>n</mml:mi><mml:mo>κ</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives>. Some applications concerning Galton–Watson trees and coalescence processes are provided.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 290‐316, 2014</p> </abstract>