{"id":357893,"date":"2017-01-25T14:28:38","date_gmt":"2017-01-25T22:28:38","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/?post_type=msr-research-item&p=357893"},"modified":"2018-10-16T20:01:58","modified_gmt":"2018-10-17T03:01:58","slug":"looping-constant-zd","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/looping-constant-zd\/","title":{"rendered":"The Looping Constant of Z^d"},"content":{"rendered":"

The looping constant \u03be<\/span>(<\/span>Z<\/span>d<\/span><\/span>)<\/span><\/span><\/span><\/span> is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in Z<\/span>d<\/span><\/span><\/span><\/span><\/span>. Poghosyan, Priezzhev and Ruelle, and independently, Kenyon and Wilson, proved recently that \u03be<\/span>(<\/span>Z<\/span>2<\/span><\/span>)<\/span>=<\/span>5<\/span>\/<\/span><\/span><\/span>4<\/span><\/span><\/span><\/span>.
\nWe consider the infinite volume limits as G<\/span>\u2191<\/span>Z<\/span>d<\/span><\/span><\/span><\/span><\/span> 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 \u03be<\/span>(<\/span>Z<\/span>d<\/span><\/span>)<\/span><\/span><\/span><\/span>. In the case of Z<\/span>2<\/span><\/span><\/span><\/span><\/span> their respective values are 8, 17\/8 and 1\/8.<\/p>\n","protected":false},"excerpt":{"rendered":"

The looping constant \u03be(Zd) is the expected number of neighbors of the origin that lie on the infinite loop-erased random walk in Zd. Poghosyan, Priezzhev and Ruelle, and independently, Kenyon and Wilson, proved recently that \u03be(Z2)=5\/4. We consider the infinite volume limits as G\u2191Zd of three different statistics: (1) The expected length of the cycle […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Cornell University 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