New methods to bound the critical probability in fractal percolation. Issue 4 (31st July 2014)

• Record Type: Journal Article
• Authors: Don, Henk
• Is Part Of: Random structures & algorithms. Volume 47:Issue 4(2015)
• Page Start: 710
• Page End: 730
• Publication Date: 2014-07-31
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20566
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>ABSTRACT</title> <p>We study the critical probability <italic>p</italic><sub><italic>c</italic></sub>(<italic>M</italic>) in two‐dimensional <italic>M</italic>‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to <italic>p</italic><sub><italic>c</italic></sub>(<italic>M</italic>). We prove that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpks" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20566:rsa20566-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn>0.881</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgphr" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20566:rsa20566-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>></mml:mo><mml:mn>0.784</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>.</p> <p>For the upper bounds, we introduce an iterative random process on a finite alphabet <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpg7" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20566:rsa20566-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math></alternatives></inline-formula>, which is easier to analyze than the original process. We show that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpd6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20566:rsa20566-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>0.993</mml:mn><mml:mo>, </mml:mo><mml:mo> </mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>0.940</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgkrnvgpb5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20566:rsa20566-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>4</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:mo>≤</mml:mo><mml:mn>0.972</mml:mn></mml:mrow></mml:math></alternatives></inline-formula>. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015</p> </abstract>