The mixing time of the giant component of a random graph1. Issue 3 (14th May 2014)

• Record Type: Journal Article
• Authors: Benjamini, Itai; Kozma, Gady; Wormald, Nicholas
• Is Part Of: Random structures & algorithms. Volume 45:Issue 3(2014)
• Page Start: 383
• Page End: 407
• Publication Date: 2014-05-14
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20539
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>We show that the total variation mixing time of the simple random walk on the giant component of supercritical <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9616h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20539:rsa20539-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v9611r" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20539:rsa20539-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>, </mml:mo><mml:mi>m</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> is <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh12v96060" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20539:rsa20539-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Θ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mrow><mml:mi>log</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. This statement was proved, independently, by Fountoulakis and Reed. Our proof follows from a structure result for these graphs which is interesting in its own right. We show that these graphs are "decorated expanders" — an expander glued to graphs whose size has constant expectation and exponential tail, and such that each vertex in the expander is glued to no more than a constant number of decorations. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 383–407, 2014</p> </abstract>