Sizes of the largest clusters for supercritical percolation on random recursive trees. Issue 1 (17th July 2012)

• Record Type: Journal Article
• Authors: Bertoin, Jean
• Is Part Of: Random structures & algorithms. Volume 44:Issue 1(2014)
• Page Start: 29
• Page End: 44
• Publication Date: 2012-07-17
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20448
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We consider Bernoulli bond‐percolation on a random recursive tree of size <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4n9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20448:rsa20448-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≫</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives>, with supercritical parameter <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4cw" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20448:rsa20448-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>t</mml:mi><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> for some <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j4jn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20448:rsa20448-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives> fixed. We show that with high probability, the largest cluster has size close to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j464" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20448:rsa20448-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi mathvariant="normal">e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:msup><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives> whereas the next largest clusters have size of order <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3z37j487" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20448:rsa20448-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>/</mml:mo><mml:mi>ln</mml:mi><mml:mo>⁡</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math></alternatives>only and are distributed according to some Poisson random measure. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 29–44, 2014</p> </abstract>