作者：Hu, Xiaoyu; Miller, Jason; Peres, Yuval
作者单位：Chinese Academy of Sciences; Stanford University; Microsoft
摘要：Let U subset of C be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product integral(U) del f(x).del g(x)dx. The set T (a; U) of a-thick points of F consists of those z is an element of U such that the average of F on a disk of radius r centered at z has growth root a/pi log 1/r as r -> 0. We show that for each 0 <= a <= 2 the Hausdorff dimension of T (a; U) is almost surely 2 - a, that nu(2-a)(T (a; ...
作者：Bollobas, Bela; Borgs, Christian; Chayes, Jennifer; Riordan, Oliver
作者单位：University of Cambridge; University of Memphis; Microsoft; University of Oxford
摘要：In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G(n)). Let lambda(n) be the largest eigenvalue of the adjacency matrix of G(n), and let G(n)(p(n)) be the random subgraph of G(n) obtained by keeping each edge independently with probability p(n). We show that the appearance of a giant component in G(n)(P-n) has a sharp threshold at p(n) = 1/lambda(n). In fact, we prove much more: if (G(n)) converges to an irreducible limit, then the density of the ...
