Interference percolation1. Issue 4 (5th March 2013)

• Record Type: Journal Article
• Authors: Balister, Paul; Bollobás, Béla
• Is Part Of: Random structures & algorithms. Volume 44:Issue 4(2014)
• Page Start: 399
• Page End: 418
• Publication Date: 2013-03-05
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20484
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Let <italic>G</italic> be an infinite connected graph with minimum degree δ and maximum degree Δ. Let <italic>G</italic><sub><italic>p</italic></sub> be a random induced subgraph of <italic>G</italic> obtained by selecting each vertex of <italic>G</italic> independently with probability <italic>p</italic>, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g06k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>0</mml:mn><mml:mo><</mml:mo><mml:mi>p</mml:mi><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives>, and let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g031" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>G</mml:mi><mml:mi>p</mml:mi><mml:mrow><mml:mo>≤</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives> be the induced subgraph of <italic>G</italic><sub><italic>p</italic></sub> obtained by deleting all vertices of <italic>G</italic><sub><italic>p</italic></sub> with degree greater than <italic>k</italic> in <italic>G</italic><sub><italic>p</italic></sub>. We show that if <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g02h" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo>≥</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g052" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Δ</mml:mo><mml:mo>/</mml:mo><mml:mo>δ</mml:mo></mml:mrow></mml:math></alternatives> is not too large then <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g04j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>G</mml:mi><mml:mi>p</mml:mi><mml:mrow><mml:mo>≤</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives> almost surely has no infinite component. Moreover, this result is essentially best possible since there are examples where <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fzxh" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>G</mml:mi><mml:mi>p</mml:mi><mml:mrow><mml:mo>≤</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives> has an infinite component (a) when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fzz1" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo>=</mml:mo><mml:mo>Δ</mml:mo><mml:mo>=</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives>, 4, or 5, and <bold><italic>k</italic></bold> = 3; (b) when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g00g" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>Δ</mml:mo><mml:mo>≫</mml:mo><mml:mo>δ</mml:mo></mml:mrow></mml:math></alternatives> for any δ and <bold><italic>k</italic></bold> = 3; and (c) when <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4g010" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo>=</mml:mo><mml:mo>Δ</mml:mo></mml:mrow></mml:math></alternatives> for any <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fwjv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>δ</mml:mo><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math></alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fwc8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:math></alternatives>. In addition, we show that if <bold><italic>G</italic> is the <italic>d</italic></bold>‐dimensional lattice <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fwds" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℤ</mml:mo><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math></alternatives> then <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh6c4fwrz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:rsa20484:rsa20484-math-0013" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>G</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>d</mml:mi></mml:mrow><mml:mrow><mml:mo>≤</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></alternatives> almost surely has an infinite component for sufficiently large <italic>d</italic>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg. 44, 399–418, 2014</p> </abstract>