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作者：Bally, Vlad; Caramellino, Lucia; Poly, Guillaume
作者单位：Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel; Centre National de la Recherche Scientifique (CNRS); University of Rome Tor Vergata; Universite de Rennes
摘要：In this article, we consider the following family of random trigonometric polynomials pn(t,Y)=Sigma k=1nYk1cos(kt)+Yk2sin(kt) for a given sequence of i.i.d. random variables Yki, i is an element of{1,2}, k >= 1, which are centered and standardized. We set N([0,pi],Y) the number of real roots over [0,pi] and N([0,pi],G) the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that lim (...
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作者：Benoist, T.; Fraas, M.; Pautrat, Y.; Pellegrini, C.
作者单位：Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Centre National de la Recherche Scientifique (CNRS); KU Leuven; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay
摘要：We study a class of Markov chains that model the evolution of a quantum system subject to repeated measurements. Each Markov chain in this class is defined by a measure on the space of matrices, and is then given by a random product of correlated matrices taken from the support of the defining measure. We give natural conditions on this support that imply that the Markov chain admits a unique invariant probability measure. We moreover prove the geometric convergence towards this invariant meas...
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作者：Imamura, Takashi; Sasamoto, Tomohiro
作者单位：Chiba University; Institute of Science Tokyo; Tokyo Institute of Technology
摘要：We consider the q-totally asymmetric simple exclusion process (q-TASEP) in the stationary regime and study the fluctuation of the position of a particle. We first observe that the problem can be studied as a limiting case of an N-particle q-TASEP with a random initial condition and with particle dependent hopping rate. Then we explain how this N-particle q-TASEP can be encoded in a dynamics on a two-sided Gelfand-Tsetlin cone described by a two-sided q-Whittaker process and present a Fredholm ...
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作者：Huang, Jiaoyang; Landon, Benjamin
作者单位：Harvard University
摘要：We study Dyson Brownian motion with general potential V and for general beta >= 1. For short times t = o(1) and under suitable conditions on V we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale eta(*) << t << 1, we prov...
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作者：Naumov, Alexey; Spokoiny, Vladimir; Ulyanov, Vladimir
作者单位：HSE University (National Research University Higher School of Economics); Russian Academy of Sciences; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; Lomonosov Moscow State University
摘要：Let X1, ... ,X-n be i.i.d. sample in Rp with zero mean and the covariance matrix Sigma. The problem of recovering the projector onto an eigenspace of Sigma from these observations naturally arises in many applications. Recent technique from Koltchinskii and Lounici (Ann Stat 45(1):121-157,2017) helps to study the asymptotic distribution of the distance in the Frobenius norm parallel to P-r-(P) over cap (r)parallel to(2) between the true projector Pr on the subspace of the rth eigenvalue and it...
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作者：Kabluchko, Zakhar; Marynych, Alexander; Temesvari, Daniel; Thaele, Christoph
作者单位：University of Munster; Ministry of Education & Science of Ukraine; Taras Shevchenko National University of Kyiv; Ruhr University Bochum
摘要：Let U1, U2,... be random points sampled uniformly and independently from the d-dimensional upper half-sphere. We showthat, as n.8, the f -vector of the (d+1)-dimensional convex cone Cn generated by U1,..., Un weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the f -vector of Cn and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of Cn can be expressed through the expected...
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作者：Hough, Robert
作者单位：State University of New York (SUNY) System; Stony Brook University
摘要：A local limit theorem is proven on connected, simply connected nilpotent Lie groups, for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization. The result extends an earlier local limit theorem of Alexopoulos which treated absolutely continuous measures with a continuous density of compact support, and also extends local limit theorems of Breuillard and Diaconis-Hough which treated general measures on the Heisenberg gr...
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作者：Gwynne, Ewain; Holden, Nina; Sun, Xin
作者单位：Massachusetts Institute of Technology (MIT); Columbia University
摘要：Let (0,2) and let h be the random distribution on C which describes a -Liouville quantum gravity (LQG) cone. Also let =16/2>4 and let be a whole-plane space-filling SLE curve sampled independent from h and parametrized by -quantum mass with respect to h. We study a family {GE}E>0 of planar maps associated with (h,) called the LQG structure graphs (a.k.a. mated-CRT maps) which we conjecture converge in probability in the scaling limit with respect to the Gromov-Hausdorff topology to a random me...
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作者：Che, Ziliang; Landon, Benjamin
作者单位：Harvard University
摘要：We consider the local statistics of H = V * XV + U * YU where V and U are independent Haar-distributed unitary matrices, and X and Y are deterministic real diagonal matrices. In the bulk, we prove that the gap statistics and correlation functions coincide with the GUE in the limit when the matrix size N -> infinity under mild assumptions on X and certain rigidity assumptions on Y (the latter being an assumption on the convergence of the eigenvalues of Y to the quantiles of its limiting spectra...
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作者：Nestoridi, Evita
作者单位：Princeton University
摘要：This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also p...
