The looping constant of ℤd. Issue 1 (13th February 2013) - Levine, Lionel; Peres, Yuval
Abstract The looping constant<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hdm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ξ(ℤd)</alternatives> is the expected number of neighbors of the origin that lie on the infinite loop‐erased random walk in <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hh5" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ℤd</alternatives>. Poghosyan, Priezzhev, and Ruelle, and independently, Kenyon and Wilson, proved recently that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hgn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ξ(ℤ2)=54</alternatives>. We consider the infinite volume limits as <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h92" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />G↑ℤd</alternatives> of three different statistics: (1) The expected length of the cycle in a uniform spanning unicycle of G; (2) The expected density of a uniform recurrent state of the abelian sandpile model on G; and (3) The ratio of the number of spanning unicycles of G to the number of rooted spanning trees of G. We show that all three limits are rational functions of the looping constant <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h8j" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ξ(ℤd)</alternatives>. In the case of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hc3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ℤ2</alternatives>, their respective values are 8, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4hbk" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />178</alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghmrs4h71" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />18</alternatives>. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 1–13, 2014