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作者：Hammond, Alan; Pete, Gabor; Schramm, Oded
作者单位：University of Oxford; Budapest University of Technology & Economics
摘要：In dynamical critical site percolation on the triangular lattice or bond percolation on Z(2), we define and study a local time measure on the exceptional times at which the origin is in an infinite cluster. We show that at a typical time with respect to this measure, the percolation configuration has the law of Kesten's incipient infinite cluster. In the most technical result of this paper, we show that, on the other hand, at the first exceptional time, the law of the configuration is differen...
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作者：Gorin, Vadim; Panova, Greta
作者单位：Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences; University of Pennsylvania
摘要：We develop a new method for studying the asymptotics of symmetric polynomials of representation-theoretic origin as the number of variables tends to infinity. Several applications of our method are presented: We prove a number of theorems concerning characters of infinite-dimensional unitary group and their q-deformations. We study the behavior of uniformly random lozenge tilings of large polygonal domains and find the GUE-eigenvalues distribution in the limit. We also investigate similar beha...
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作者：Perkowski, Nicolas; Ruf, Johannes
作者单位：Universite PSL; Universite Paris-Dauphine; University of London; University College London
摘要：Certain countably and finitely additive measures can be associated to a given nonnegative supermartingale. Under weak assumptions on the underlying probability space, existence and (non)uniqueness results for such measures are proven.
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作者：Panchenko, Dmitry
作者单位：University of Toronto
摘要：The authors of [Ann. Henri Poincare 16 (2015) 691-708] introduced a multi-species version of the Sherrington-Kirkpatrick model and suggested the analogue of the Parisi formula for the free energy. Using a variant of Guerra's replica symmetry breaking interpolation, they showed that, under certain assumption on the interactions, the formula gives an upper bound on the limit of the free energy. In this paper we prove that the bound is sharp. This is achieved by developing a new multi-species for...
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作者：Kiss, Demeter; Manolescu, Ioan; Sidoravicius, Vladas
作者单位：University of Cambridge; University of Geneva; Instituto Nacional de Matematica Pura e Aplicada (IMPA)
摘要：Self-destructive percolation with parameters p, delta is obtained by taking a site percolation configuration with parameter p, closing all sites belonging to infinite clusters, then opening every closed site with probability delta, independently of the rest. Call theta (p, delta) the probability that the origin is in an infinite cluster in the configuration thus obtained. For two-dimensional lattices, we show the existence of delta > 0 such that, for any p > p(c), theta (p, delta) = 0. This pr...
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作者：Chen, Le; Dalang, Robert C.
作者单位：Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
摘要：We study the nonlinear stochastic heat equation in the spatial domain R, driven by space-time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall's lemma, by keeping tight control over m...
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作者：Lubetzky, Eyal; Steif, Jeffrey E.
作者单位：Chalmers University of Technology; University of Gothenburg
摘要：The noise sensitivity of a Boolean function describes its likelihood to flip under small perturbations of its input. Introduced in the seminal work of Benjamini, Kalai and Schramm [Inst. Hautes Etudes Sci. Publ. Math. 90 (1999) 5-43], it was there shown to be governed by the first level of Fourier coefficients in the central case of monotone functions at a constant critical probability p(c). Here we study noise sensitivity and a natural stronger version of it, addressing the effect of noise gi...
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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：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...
