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作者：Hammond, Alan
作者单位：University of Oxford
摘要：We study the droplet that results from conditioning the planar subcritical Fortuin-Kasteleyn random cluster model on the presence of an open circuit Gamma(0) encircling the origin and enclosing an area of at least (or exactly) n(2). We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Gamma(0)), this being the maximum distance from a point in the circuit Gamma(0) to the boundary a conv(Gamma(0)) of the circuit's convex hull; and in...
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作者：Hu, Yaozhong; Lu, Fei; Nualarti, David
作者单位：University of Kansas
摘要：In this paper, a Feynman-Kac formula is established for stochastic partial differential equation driven by Gaussian noise which is, with respect to time, a fractional Brownian motion with Hurst parameter H < 1/2. To establish such a formula, we introduce and study a nonlinear stochastic integral from the given Gaussian noise. To show the Feynman-Kac integral exists, one still needs to show the exponential integrability of nonlinear stochastic integral. Then, the approach of approximation with ...
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作者：Tao, Terence; Vu, Van
作者单位：University of California System; University of California Los Angeles; Rutgers University System; Rutgers University New Brunswick
摘要：We study the eigenvalues of the covariance matrix 1/n M*M of a large rectangular matrix M = M-n,M-p = (zeta(ij))(1 <= i <= p;1 <= j <= n) whose entries are i.i.d. random variables of mean zero, variance one, and having finite C(0)th moment for some sufficiently large constant C-0. The main result of this paper is a Four Moment theorem for i.i.d. covariance matrices (analogous to the Four Moment theorem for Wigner matrices established by the authors in [Acta Math. (2011) Random matrices: Univer...
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作者：Damron, Michael; Sapozhnikov, Artem
作者单位：Princeton University; Swiss Federal Institutes of Technology Domain; ETH Zurich
摘要：We prove limit theorems and variance estimates for quantities related to ponds and outlets for 2D invasion percolation. We first exhibit several properties of a sequence (O(n)) of outlet variables, the nth of which gives the number of outlets in the box centered at the origin of side length 2(n). The most important of these properties describes the sequence's renewal structure and exponentially fast mixing behavior. We use these to prove a central limit theorem and strong law of large numbers ...
