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作者：Kenyon, Richard
作者单位：Brown University
摘要：The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process...
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作者：Haggstrom, Olle
作者单位：Chalmers University of Technology
摘要：Oded Schramm (1961-2008) influenced greatly the development of percolation theory beyond the usual Z(d) setting; in particular, the case of nonamenable lattices. Here, we review some of his work in this field.
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作者：Schramm, Oded; Smirnov, Stanislav; Garban, Christophe
作者单位：University of Geneva; Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS)
摘要：We prove Tsirelson's conjecture that any scaling limit of the critical planar percolation is a black noise. Our theorems apply to a number of percolation models, including site percolation on the triangular grid and any subsequential scaling limit of bond percolation on the square grid. We also suggest a natural construction for the scaling limit of planar percolation, and more generally of any discrete planar model describing connectivity properties.
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作者：Berestycki, Nathanaeel; Schramm, Oded; Zeitouni, Ofer
作者单位：University of Cambridge; Weizmann Institute of Science; University of Minnesota System; University of Minnesota Twin Cities
摘要：Let S(n) be the permutation group on n elements, and consider a random walk on S(n) whose step distribution is uniform on k-cycles. We prove a well-known conjecture that the mixing time of this process is (1/k)n log n, with threshold of width linear in n. Our proofs are elementary and purely probabilistic, and do not appeal to the representation theory of S(n).
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作者：Blachere, Sebastien; den Hollander, Frank; Steif, Jeffrey E.
作者单位：Aix-Marseille Universite; Leiden University; Leiden University - Excl LUMC; Chalmers University of Technology; University of Gothenburg
摘要：In this paper, we consider a random walk and a random color scenery on Z. The increments of the walk and the colors of the scenery are assumed to be i.i.d. and to be independent of each other. We are interested in the random process of colors seen by the walk in the course of time. Bad configurations for this random process are the discontinuity points of the conditional probability distribution for the color seen at time zero given the colors seen at all later times. We focus on the case wher...
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作者：Sheffield, Scott; Wilson, David B.
作者单位：Massachusetts Institute of Technology (MIT); Microsoft
摘要：Gerard Watts predicted a formula for the probability in percolation that there is both a left-right and an up-down crossing, which was later proved by Julien Dubedat. Here we present a simpler proof due to Oded Schramm, which builds on Cardy's formula in a conceptually appealing way: the triple derivative of Cardy's formula is the sum of two multi-arm densities. The relative sizes of the two terms are computed with Girsanov conditioning. The triple integral of one of the terms is equivalent to...
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作者：Garban, Christophe
作者单位：Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS)
摘要：We survey in this paper the main contributions of Oded Schramm related to noise sensitivity. We will describe in particular his various works which focused on the spectral analysis of critical percolation (and more generally of Boolean functions), his work on the shape-fluctuations of first passage percolation and finally his contributions to the model of dynamical percolation.
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作者：Lawler, Gregory F.; Sheffield, Scott
作者单位：University of Chicago; University of Chicago; Massachusetts Institute of Technology (MIT)
摘要：The Schramm-Loewner evolution (SLE kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When kappa < 8, an instance of SLE kappa is a random planar curve with almost sure Hausdorff dimension d = 1 + kappa/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume. For kappa < 8, we use a Doob-Meyer decomposition to construct the unique (under mild assumptions) Markovi...
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作者：Kozma, Gady
作者单位：Weizmann Institute of Science
摘要：We show that critical percolation on a product of two regular trees of degree >= 3 satisfies the triangle condition. The proof does not examine the degrees of vertices and is not perturbative in any sense. It relies on an unpublished lemma of Oded Schramm.
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作者：Holroyd, Alexander E.; Lyons, Russell; Soo, Terry
作者单位：Microsoft; Indiana University System; Indiana University Bloomington; University of Victoria
摘要：Given a homogeneous Poisson process on R-d with intensity lambda, we prove that it is possible to partition the points into two sets, as a deterministic function of the process, and in an isometry-equivariant way, so that each set of points forms a homogeneous Poisson process, with any given pair of intensities summing to lambda. In particular, this answers a question of Ball [Electron. Commun. Probab. 10 (2005) 60-69], who proved that in d = 1, the Poisson points may be similarly partitioned ...
