乐胖代购免代理版

{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,6,12]],"date-time":"2024-06-12T06:40:10Z","timestamp":1718174410450},"reference-count":42,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T00:00:00Z","timestamp":1564963200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Combinator. Probab. Comp."],"published-print":{"date-parts":[[2020,1]]},"abstract":"Abstract<\/jats:title>The Hamming graphH<\/jats:italic>(d<\/jats:italic>,n<\/jats:italic>) is the Cartesian product ofd<\/jats:italic>complete graphs onn<\/jats:italic>vertices. Let

${m=d(n-1)}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be the degree and

$V = n^d$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be the number of vertices ofH<\/jats:italic>(d<\/jats:italic>,n<\/jats:italic>). Let

$p_c^{(d)}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>be the critical point for bond percolation onH<\/jats:italic>(d<\/jats:italic>,n<\/jats:italic>). We show that, for

$d \\in \\mathbb{N}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>fixed and

$n \\to \\infty$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,

$$p_c^{(d)} = {1 \\over m} + {{2{d^2} - 1} \\over {2{{(d - 1)}^2}}}{1 \\over {{m^2}}} + O({m^{ - 3}}) + O({m^{ - 1}}{V^{ - 1\/3}}),$$<\/jats:tex-math><\/jats:alternatives><\/jats:disp-formula>which extends the asymptotics found in [10] by one order. The term

$O(m^{-1}V^{-1\/3})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>is the width of the critical window. For

$d=4,5,6$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>we have

$m^{-3} = O(m^{-1}V^{-1\/3})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and so the above formula represents the full asymptotic expansion of

$p_c^{(d)}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In [16] we show that this formula is a crucial ingredient in the study of critical bond percolation onH<\/jats:italic>(d<\/jats:italic>,n<\/jats:italic>) for

$d=2,3,4$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erd\u00f6s\u2013R\u00e9nyi random graph.<\/jats:p>","DOI":"10.1017\/s0963548319000208","type":"journal-article","created":{"date-parts":[[2019,8,5]],"date-time":"2019-08-05T09:36:44Z","timestamp":1564997804000},"page":"68-100","source":"Crossref","is-referenced-by-count":3,"title":["Expansion of Percolation Critical Points for Hamming Graphs"],"prefix":"10.1017","volume":"29","author":[{"given":"Lorenzo","family":"Federico","sequence":"first","affiliation":[]},{"given":"Remco","family":"Van Der Hofstad","sequence":"additional","affiliation":[]},{"given":"Frank","family":"Den Hollander","sequence":"additional","affiliation":[]},{"given":"Tim","family":"Hulshof","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,8,5]]},"reference":[{"key":"S0963548319000208_ref28","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20074"},{"key":"S0963548319000208_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-012-0684-6"},{"key":"S0963548319000208_ref16","unstructured":"[16] Federico, L. , van der Hofstad, R. , den Hollander, F. and Hulshof, T. 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