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作者：Neininger, Ralph; Sulzbach, Henning
作者单位：Goethe University Frankfurt
摘要：Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space C[0, 1] of continuous functions endowed with uniform topology and the space D[0, 1] of cadlag functions with the Skorokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characterist...
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作者：Janson, Svante; Stefansson, Sigurdur Orn
作者单位：Uppsala University; Royal Institute of Technology; Nordic Institute for Theoretical Physics; Stockholm University
摘要：We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges n of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by n(-1/2) is described by a Brownian excursion. The planar maps, with the graph met...
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作者：Khoshnevis, Davar; Xiao, Yimin
作者单位：Utah System of Higher Education; University of Utah; Michigan State University
摘要：Let W denote d-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of W (E) boolean AND F, where E subset of (0, infinity) and F subset of R-d are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when d >= 2, our formula can be described in terms of the Hausdorff dimension of E x F, where E x F is viewed as a subspace of ...
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作者：Cox, A. M. G.; Peskir, G.
作者单位：University of Bath; University of Manchester
摘要：We present a constructive probabilistic proof of the fact that if B = (B-t)(t >= 0) is standard Brownian motion started at 0, and mu is a given probability measure on R such that mu({0}) = 0, then there exists a unique left-continuous increasing function b: (0, infinity) -> R boolean OR {+infinity} and a unique left-continuous decreasing function c: (0, infinity) -> R boolean OR (-infinity) such that B stopped at tau(b,c) = inf{t > 0 vertical bar B-t >= b(t) or B-t <= c(t)} has the law p,. The...
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作者：Jiang, Tiefeng; Matsumoto, Sho
作者单位：University of Minnesota System; University of Minnesota Twin Cities; Nagoya University
摘要：Let theta(1) ,..., theta(n) be random variables from Dyson's circular beta-ensemble with probability density function Const. Pi(1 <= j<= n) vertical bar e(i theta j) - e(i theta k)vertical bar(beta). For each n >= 2 and beta > 0, we obtain some inequalities on E[p(mu)(Z(n))<(P-v(Z(n)))over bar>], where Z(n) = (e(i theta 1) ,..., e(i theta n)) and p(mu) is the power-sum symmetric function for partition mu. When beta = 2, our inequalities recover an identity by Diaconis and Evans for Haar-invari...
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作者：Cass, Thomas; Hairer, Martin; Litterer, Christian; Tindel, Samy
作者单位：Imperial College London; University of Warwick; Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Universite de Lorraine
摘要：We consider stochastic differential equations of the form dY(t) = V(Y-t)dX(t) + V-0(Y-t)dt driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields V-0 and V = (V-1,...,V-d) satisfy Hormander's bracket condition, we demonstrate that Y-t admits a smooth density for any t is an element of (0, T], provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the ...
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作者：Eldan, Ronen
作者单位：Microsoft
摘要：We consider an analogous version of the diffusion-limited aggregation model defined on the hyperbolic plane. We prove that almost surely the aggregate viewed at time infinity will have a positive density.
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作者：Baur, Erich; Bolthausen, Erwin
作者单位：University of Zurich
摘要：We study exit laws from large balls in Z(d), d >= 3, of random walks in an i.i.d. random environment that is a small perturbation of the environment corresponding to simple random walk. Under a centering condition on the measure governing the environment, we prove that the exit laws are close to those of a symmetric random walk, which we identify as a perturbed simple random walk. We obtain bounds on total variation distances as well as local results comparing exit probabilities on boundary se...
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作者：Ding, Jian; Eldan, Ronen; Zhai, Alex
作者单位：University of Chicago; Stanford University
摘要：We prove two theorems concerning extreme values of general Gaussian fields. Our first theorem concerns the phenomenon of multiple peaks. Consider a centered Gaussian field whose sites have variance at most 1, and let rho be the standard deviation of its supremum. A theorem of Chatterjee states that when such a Gaussian field is superconcentrated (i.e., rho << 1), it typically attains values near its maximum on multiple almost-orthogonal sites and is said to exhibit multiple peaks. We improve h...
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作者：Benjamini, Itai; Duminil-Copin, Hugo; Kozma, Gady; Yadin, Ariel
作者单位：Weizmann Institute of Science; University of Geneva; Ben-Gurion University of the Negev
摘要：We study harmonic functions on random environments with particular emphasis on the case of the infinite cluster of supercritical percolation on Z(d). We prove that the vector space of harmonic functions growing at most linearly is (d + 1)-dimensional almost surely. Further, there are no nonconstant sublinear harmonic functions (thus implying the uniqueness of the corrector). A main ingredient of the proof is a quantitative, annealed version of the Avez entropy argument. This also provides boun...
