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作者：Rakhlin, Alexander; Sridharan, Karthik; Tewari, Ambuj
作者单位：University of Pennsylvania; University of Michigan System; University of Michigan
摘要：We establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide sequential (non-i.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform conv...
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作者：Cacciapuoti, Claudio; Maltsev, Anna; Schlein, Benjamin
作者单位：University of Bonn; University of Bristol
摘要：We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from ErdAs et al. (Adv Math 229(3):1435-1515, 2012; Electron J Probab 18(59):1-58, 2013), by removing the logarithmic corrections. As applications...
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作者：Calka, Pierre; Yukich, J. E.
作者单位：Universite de Rouen Normandie; Lehigh University
摘要：Let K-n be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on . We establish variance asymptotics as for the re-scaled intrinsic volumes and -face functionals of , , resolving an open problem (Weil and Wieacker, Handbook of Convex Geometry, vol. B, pp. 1391-1438. North-Holland/Elsevier, Amsterdam, 1993). Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process ...
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作者：Aizenman, Michael; Warzel, Simone
作者单位：Princeton University; Princeton University; Technical University of Munich
摘要：We consider the distribution of the values at real points of random functions which belong to the Herglotz-Pick (HP) class of analytic mappings of the upper half plane into itself. It is shown that under mild stationarity assumptions the individual values of HP functions with singular spectra have a Cauchy type distribution. The statement applies to the diagonal matrix elements of random operators, and holds regardless of the presence or not of level repulsion, i.e. applies to both random matr...
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作者：Brzezniak, Z.; Cerrai, S.; Freidlin, M.
作者单位：University of York - UK; University System of Maryland; University of Maryland College Park
摘要：We are dealing with the Navier-Stokes equation in a bounded regular domain of , perturbed by an additive Gaussian noise , which is white in time and colored in space. We assume that the correlation radius of the noise gets smaller and smaller as , so that the noise converges to the white noise in space and time. For every we introduce the large deviation action functional and the corresponding quasi-potential and, by using arguments from relaxation and -convergence we show that converges to , ...
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作者：Cai, Tony; Ma, Zongming; Wu, Yihong
作者单位：University of Pennsylvania; University of Illinois System; University of Illinois Urbana-Champaign
摘要：This paper considers a sparse spiked covariance matrix model in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established, which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices. We...
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作者：Hammond, Alan
作者单位：University of Oxford
摘要：We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists such that, for any , the set of parameter values at which the random stirring model on the rooted regular tree with offspring degree almost surely contains an infinite cycle consists of a semi-infinite interval. The critical point at the left-hand end of this interval is at least and at most .
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作者：Mathieu, Pierre; Mourrat, Jean-Christophe
作者单位：Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne; Centre National de la Recherche Scientifique (CNRS); Ecole Normale Superieure de Lyon (ENS de LYON)
摘要：We show aging of Glauber-type dynamics on the random energy model, in the sense that we obtain the annealed scaling limits of the clock process and of the age process. The latter encodes the Gibbs weight of the configuration occupied by the dynamics. Both limits are expressed in terms of stable subordinators.
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作者：Debussche, A.; Vovelle, J.
作者单位：Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet
摘要：Under an hypothesis of non-degeneracy of the flux, we study the long-time behaviour of periodic scalar first-order conservation laws with stochastic forcing in any space dimension. For sub-cubic fluxes, we show the existence of an invariant measure. Moreover for sub-quadratic fluxes we show uniqueness and ergodicity of the invariant measure. Also, since this invariant measure is supported by for some small, we are led to generalize to the stochastic case the theory of solutions developed by Ch...
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作者：Junge, Marius; Zeng, Qiang
作者单位：University of Illinois System; University of Illinois Urbana-Champaign
摘要：We prove a deviation inequality for noncommutative martingales by extending Oliveira's argument for random matrices. By integration we obtain a Burkholder type inequality with satisfactory constant. Using continuous time, we establish noncommutative Poincar, type inequalities for nice semigroups with a positive curvature condition. These results allow us to prove a general deviation inequality and a noncommutative transportation inequality due to Bobkov and Gotze in the commutative case. To de...
