The time of bootstrap percolation with dense initial sets for all thresholds1. Issue 1 (7th April 2014)

• Record Type: Journal Article
• Authors: Bollobas, Béla; Smith, Paul; Uzzell, Andrew J.
• Is Part Of: Random structures & algorithms. Volume 47:Issue 1(2015)
• Page Start: 1
• Page End: 29
• Publication Date: 2014-04-07
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20529
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>We study the percolation time of the <italic>r</italic>‐neighbour bootstrap percolation model on the discrete torus <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqhph9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20529:rsa20529-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ℤ</mml:mi><mml:mo>/</mml:mo><mml:mi>n</mml:mi><mml:mi>ℤ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>. For <italic>t</italic> at most a polylog function of <italic>n</italic> and initial infection probabilities within certain ranges depending on <italic>t</italic>, we prove that the percolation time of a random subset of the torus is exactly equal to <italic>t</italic> with high probability as <italic>n</italic> tends to infinity. Our proof rests crucially on three new extremal theorems that together establish an almost complete understanding of the geometric behaviour of the <italic>r</italic>‐neighbour bootstrap process in the dense setting. The special case <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj14cqhp6b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20529:rsa20529-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:mo>−</mml:mo><mml:mi>r</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> of our result was proved recently by Bollobas, Holmgren, Smith and Uzzell. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 1–29, 2015</p> </abstract>