When does the top homology of a random simplicial complex vanish?1. Issue 1 (13th March 2013)

• Record Type: Journal Article
• Authors: Aronshtam, Lior; Linial, Nathan
• Is Part Of: Random structures & algorithms. Volume 46:Issue 1(2015)
• Page Start: 26
• Page End: 35
• Publication Date: 2013-03-13
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20495
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>Several years ago Linial and Meshulam (Combinatorica 26 (2006) 457–487) introduced a model called <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rd4f8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20495:rsa20495-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mi>d</mml:mi></mml:msub><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> of random <italic>n</italic>‐vertex <italic>d</italic>‐dimensional simplicial complexes. The following question suggests itself very naturally: What is the threshold probability <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgh2d2rd4x0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20495:rsa20495-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> at which the <italic>d</italic>‐dimensional homology of such a random <italic>d</italic>‐complex is, almost surely, nonzero? Here we derive an upper bound on this threshold. Computer experiments that we have conducted suggest that this bound may coincide with the actual threshold, but this remains an open question. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 26–35, 2015</p> </abstract>