1,721,072 research outputs found
A diffusive matrix model for invariant β-ensembles
Full text linkWe define a new diffusive matrix model converging towards the β -Dyson Brownian motion for all β ∈ [0 , 2] that provides an explicit construction of β -ensembles of ran- dom matrices that is invariant under the orthogonal/unitary group. We also describe the eigenvector dynamics of the limiting matrix process; we show that when β < 1 and that two eigenvalues collide, the eigenvectors of these two colliding eigenval- ues fluctuate very fast and take the uniform measure on the orthocomplement of the eigenvectors of the remaining eigenvalues. Keywords: random matrices; stochastic calculus; Interacting particles system; Dyson Brownian motion
Convergence of the spectral measure of non-normal matrices
No full textWe discuss regularization by noise of the spectrum of large random non-normal matrices. Under suitable conditions, we show that the regularization of a sequence of matrices that converges in *-moments to a regular element a by the addition of a polynomially vanishing Gaussian Ginibre matrix forces the empirical measure of eigenvalues to converge to the Brown measure of a.France. Agence nationale de la recherche (Project ANR-08-BLAN-0311-01
Transport Maps for β-Matrix Models and Universality
No full textWe construct approximate transport maps for non-critical β-matrix models, that is, maps so that the push forward of a non-critical β-matrix model with a given potential is a non-critical β-matrix model with another potential, up to a small error in the total variation distance. One of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models.National Science Foundation (U.S.) (Grant DMS-1262411)National Science Foundation (U.S.) (Grant DMS-1307704)Simons Foundatio
Beyond universality in random matrix theory
No full textIn order to have a better understanding of finite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices
Free monotone transport
Full text linkBy solving a free analog of the Monge-Ampère equation, we prove a non-commutative analog of Brenier’s monotone transport theorem: if an n-tuple of self-adjoint non-commutative random variables Z [subscript 1],…,Z [subscript n] satisfies a regularity condition (its conjugate variables ξ [subscript 1],…,ξ [subscript n] should be analytic in Z [subscript 1],…,Z [subscript n] and ξ[subscript j] should be close to Z [subscript j] in a certain analytic norm), then there exist invertible non-commutative functions F [subscript j] of an n-tuple of semicircular variables S [subscript 1],…,S [subscript n], so that Z [subscript j] =F [subscript j] (S [subscript 1],…,S [subscript n] ). Moreover, F [subscript j] can be chosen to be monotone, in the sense that and g is a non-commutative function with a positive definite Hessian. In particular, we can deduce that C[superscript ∗](Z[subscript 1],…,Z [subscript n] )≅C[superscript ∗](S [subscript 1],…,S [subscript n] ) and W[superscript ∗](Z[subscript 1],…,Z[subscript n])≅L(F(n)) . Thus our condition is a useful way to recognize when an n-tuple of operators generate a free group factor. We obtain as a consequence that the q-deformed free group factors Γ[subscript q](R[superscript n]) are isomorphic (for sufficiently small q, with bound depending on n) to free group factors. We also partially prove a conjecture of Voiculescu by showing that free Gibbs states which are small perturbations of a semicircle law generate free group factors. Lastly, we show that entrywise monotone transport maps for certain Gibbs measure on matrices are well-approximated by the matricial transport maps given by free monotone transport.France. Agence nationale de la recherche (ANR-08-BLAN-0311-01)Simons FoundationNational Science Foundation (U.S.) (NSF grant DMS-0900776)National Science Foundation (U.S.) (Grant DMS-1161411)United States. Defense Advanced Research Projects Agency (DARPA HR0011-12-1-0009
Free analysis and random matrices
Full text linkInternational audienceWe describe the Schwinger-Dyson equation related with the free difference quotient. Such an equation appears in different fields such as combinatorics (via the problem of the enumeration of planar maps), operator algebra (via the definition of a natural integration by parts in free probability), in classical probability (via random matrices or particles in repulsive interaction). In these lecture notes, we shall discuss when this equation uniquely defines the system and in such a case how it leads to deep properties of the solution. This analysis can be extended to systems which approximately satisfy these equations, such as random matrices or Coulomb gas interacting particle systems
Asymptotics of random matrices and related models: the uses of Dyson-Schwinger equations
No full textProbability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
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