The looping constant of ℤd. Issue 1 (13th February 2013)

• Record Type: Journal Article
• Authors: Levine, Lionel; Peres, Yuval
• Is Part Of: Random structures & algorithms. Volume 45:Issue 1(2014)
• Page Start: 1
• Page End: 13
• Publication Date: 2013-02-13
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20478
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>The <italic>looping constant</italic><inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hdm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ξ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mo>ℤ</mml:mo><mml:mi>d</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> is the expected number of neighbors of the origin that lie on the infinite loop‐erased random walk in <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hh5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0003" 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>. Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hgn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ξ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mo>ℤ</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mfrac><mml:mn>5</mml:mn><mml:mn>4</mml:mn></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>. We consider the infinite volume limits as <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h92" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</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> of three different statistics: (1) The expected length of the cycle in a uniform spanning unicycle of <italic>G</italic>; (2) The expected density of a uniform recurrent state of the abelian sandpile model on <italic>G</italic>; and (3) The ratio of the number of spanning unicycles of <italic>G</italic> to the number of rooted spanning trees of <italic>G</italic>. We show that all three limits are rational functions of the looping constant <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h8j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>ξ</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mo>ℤ</mml:mo><mml:mi>d</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. In the case of <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hc3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mo>ℤ</mml:mo><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math></alternatives></inline-formula>, their respective values are 8, <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hbk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfrac><mml:mrow><mml:mn>17</mml:mn></mml:mrow><mml:mn>8</mml:mn></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h71" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:10429832:media:rsa20478:rsa20478-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>8</mml:mn></mml:mfrac></mml:mrow></mml:math></alternatives></inline-formula>. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014</p> </abstract>