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作者：Narayanan, Hariharan; Sheffield, Scott
作者单位：Tata Institute of Fundamental Research (TIFR); Massachusetts Institute of Technology (MIT)
摘要：Suppose a, beta are Lipschitz, strongly concave functions from [0, 1] to R and y is a concave function from [0, 1] to R such that a(0) = y(0) = 0, a(1)=8(0) = 0 and 8(1)=y(1) = 0. For an nxn Hermitian matrix W, let spec(W) denote the vector in R whose coordinates are the eigenvalues of W listed in nonincreasing order. Let lambda=aa, mu=-beta on (0, 1) and v=ay, at all points of (0, 1], where a is the left derivative. Let lambda(i);= n2(a)-a()), for i = [n], and similarly, () == n2(B()-B(2)) an...
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作者：Muirhead, Stephen
作者单位：University of Melbourne
摘要：We establish the sharpness of the phase transition for a wide class of Gaussian percolation models, on Z(d) or R-d, d >= 2, with correlations decaying at least algebraically with exponent alpha > 0, including the discrete Gaussian free field (d >= 3, alpha = d - 2), the discrete Gaussian membrane model (d >= 5, alpha = d - 4), and many other examples both discrete and continuous. In particular, we do not assume positive correlations. This result is new for all strongly correlated models (i.e.,...
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作者：Xu, Changji; Yang, Fan; Yau, Horng-Tzer; Yin, Jun
作者单位：Harvard University; Tsinghua University; Harvard University; University of California System; University of California Los Angeles
摘要：We consider Hermitian random band matrices H = (h(xy)) on the d-dimensional lattice (Z/LZ)(d), where the entries h(xy) = (h) over bar (yx) are independent centered complex Gaussian random variables with variances s(xy) = E|h(xy)|(2). The variance matrix S = (s(xy)) has a banded profile so that sxy is negligible if |x - y| exceeds the band width W. For dimensions d >= 7, we prove the bulk eigenvalue universality of H under the condition W >> L95/(d+95). Assuming that W >= L-epsilon for a small ...
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作者：Burdzy, Krzysztof; Englander, Janos
作者单位：University of Washington; University of Washington Seattle; University of Colorado System; University of Colorado Boulder
摘要：We show that the spine of the Fleming-Viot process driven by Brownian motion in a bounded Lipschitz domain with Lipschitz constant less than 1 converges to Brownian motion conditioned to stay in the domain forever.
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作者：Salez, Justin
作者单位：Universite PSL; Universite Paris-Dauphine
摘要：We prove that the absolute spectral gap of any monotone Markov chain coincides with its optimal Ollivier-Ricci curvature, where the word optimal refers to the choice of the underlying metric. Moreover, we provide a new expression in terms of local variations of increasing functions, which has several practical advantages over the traditional variational formulation using the Dirichlet form. As an illustration, we explicitly determine the optimal curvature and spectral gap of the nonconservativ...
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作者：Chen, Xin; Wang, Jian
作者单位：Shanghai Jiao Tong University; Fujian Normal University
摘要：Consider the Schrodinger operator L-V = - Delta + V on R-d , where V : R-d -> [0 , infinity) is a nonnegative and locally bounded potential on R-d so that for all x is an element of R-d with | x | >= 1, c(1)g( | x | ) <= V (x) <= c(2)g( | x | ) with some constants c 1 , c(2) > 0 and a nondecreasing and strictly positive function g : [0,infinity) -> [1 ,+infinity ) that satisfies g(2r) <= c(0)g(r) for all r > 0 and lim(r ->infinity )g(r) = infinity . We establish global in time and qualitativel...
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作者：Berestycki, Julien; Kim, Yujin h.; Lubetzky, Eyal; Mallein, Bastien; Zeitouni, Ofer
作者单位：University of Oxford; University of Oxford; New York University; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
摘要：We consider a branching Brownian motion in Rd with d >= 1 in which the position X(u) t is an element of Rd of a particle u at time t can be encoded by its direction theta(u) t is an element of Sd-1 and its distance R(u) t to 0. We prove that the extremal point process Sigma delta(theta(u) t,Rt(u)-m(d) t ) (where the sum is over all particles alive at time t and m(d) t is an explicit centering term) converges in distribution to a randomly shifted, decorated Poisson point process on Sd-1 x R. Mo...
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作者：Elentano, Michael
作者单位：University of California System; University of California Berkeley
摘要：We develop an approach for studying the local convexity of a certain class of random objectives around the iterates of an AMP algorithm. Our approach involves applying the Sudakov-Fernique inequality conditionally on a long sequence of AMP iterates, and our main contribution is to demonstrate the way in which the resulting objective can be simplified and analyzed. As a consequence, we provide a new, and arguably simpler, proof of some of the results of Celentano, Fan and Mei ( Ann. Statist. 51...
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作者：Herry, Ronan; Malicet, Dominique; Poly, Guillaume
作者单位：Universite de Rennes; Universite Gustave-Eiffel; Universite Paris-Est-Creteil-Val-de-Marne (UPEC)
摘要：We establish, in full generality, an unexpected phenomenon of strong regularization along normal convergence on Wiener chaoses. Namely, for every sequence of chaotic random variables, convergence in law to the Gaussian distribution is automatically upgraded to superconvergence : the regularity of the densities increases along the convergence, and all the derivatives converge uniformly on the real line. Our findings strikingly strengthen known results regarding modes of convergence for normal a...
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作者：Beliaev, Dmitry; Mcauley, Michael; Muirhead, Stephen
作者单位：University of Oxford; University of Helsinki; University of Melbourne
摘要：3 School of Mathematics and Statistics, University of Melbourne, c smui@unimelb.edu.au For a smooth stationary Gaussian field f on R d and level is an element of R, we consider the number of connected components of the excursion set { f >= } (or level set { f = } ) contained in large domains. The mean of this quantity is known to scale like the volume of the domain under general assumptions on the field. We prove that, assuming sufficient decay of correlations (e.g., the Bargmann-Fock field), ...
