First passage percolation and escape strategies1. Issue 3 (23rd May 2014)

• Record Type: Journal Article
• Authors: Andjel, Enrique D.; Vares, Maria E.
• Is Part Of: Random structures & algorithms. Volume 47:Issue 3(2015)
• Page Start: 414
• Page End: 423
• Publication Date: 2014-05-23
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20548
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Consider first passage percolation on <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgrq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0001" 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></inline-formula> with passage times given by i.i.d. random variables with common distribution <italic>F</italic>. Let <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgs8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>π</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>, </mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> be the time from <italic>u</italic> to <italic>v</italic> for a path <italic>π</italic> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgkz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>t</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>u</mml:mi><mml:mo>, </mml:mo><mml:mi>v</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> the minimal time among all paths from <italic>u</italic> to <italic>v</italic>. We ask whether or not there exist points <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgn2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:mi>y</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mo>ℤ</mml:mo><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math></alternatives></inline-formula> and a semi‐infinite path <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trgg9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>π</mml:mo><mml:mo>=</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mo>, </mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>, </mml:mo><mml:mo>…</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> such that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj2f9trghv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20548:rsa20548-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mo>π</mml:mo></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>y</mml:mi><mml:mo>, </mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo><</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo>, </mml:mo><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> for all <italic>n</italic>. Necessary and sufficient conditions on <italic>F</italic> are given for this to occur. When the support of <italic>F</italic> is unbounded, we also obtain results on the number of edges with large passage time used by geodesics. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 414–423, 2015</p> </abstract>