1,737 research outputs found
Orthogonal polynomial ensembles in probability theory
We survey a number of models from physics, statistical mechanics, probability
theory and combinatorics, which are each described in terms of an orthogonal
polynomial ensemble. The most prominent example is apparently the Hermite
ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE),
and other well-known ensembles known in random matrix theory like the Laguerre
ensemble for the spectrum of Wishart matrices. In recent years, a number of
further interesting models were found to lead to orthogonal polynomial
ensembles, among which the corner growth model, directed last passage
percolation, the PNG droplet, non-colliding random processes, the length of the
longest increasing subsequence of a random permutation, and others. Much
attention has been paid to universal classes of asymptotic behaviors of these
models in the limit of large particle numbers, in particular the spacings
between the particles and the fluctuation behavior of the largest particle.
Computer simulations suggest that the connections go even farther and also
comprise the zeros of the Riemann zeta function. The existing proofs require a
substantial technical machinery and heavy tools from various parts of
mathematics, in particular complex analysis, combinatorics and variational
analysis. Particularly in the last decade, a number of fine results have been
achieved, but it is obvious that a comprehensive and thorough understanding of
the matter is still lacking. Hence, it seems an appropriate time to provide a
surveying text on this research area.Comment: Published at http://dx.doi.org/10.1214/154957805100000177 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Eigenvalue order statistics for random Schr\"odinger operators with doubly-exponential tails
We consider random Schr\"odinger operators of the form , where
is the lattice Laplacian on and is an i.i.d.
random field, and study the extreme order statistics of the eigenvalues for
this operator restricted to large but finite subsets of . We show
that for with a doubly-exponential type of upper tail, the upper extreme
order statistics of the eigenvalues falls into the Gumbel max-order class. The
corresponding eigenfunctions are exponentially localized in regions where
takes large, and properly arranged, values. A new and self-contained argument
is thus provided for Anderson localization at the spectral edge which permits a
rather explicit description of the shape of the potential and the
eigenfunctions. Our study serves as an input into the analysis of an associated
parabolic Anderson problem.Comment: 36 page
Mean-field interaction of Brownian occupation measures, I: uniform tube property of the Coulomb functional
We study the transformed path measure arising from the self-interaction of a
three-dimensional Brownian motion via an exponential tilt with the Coulomb
energy of the occupation measures of the motion by time . The logarithmic
asymptotics of the partition function were identified in the 1980s by Donsker
and Varadhan [DV83-P] in terms of a variational formula. Recently [MV14] a new
technique for studying the path measure itself was introduced, which allows for
proving that the normalized occupation measure asymptotically concentrates
around the set of all maximizers of the formula. In the present paper, we show
that likewise the Coulomb functional of the occupation measure concentrates
around the set of corresponding Coulomb functionals of the maximizers in the
uniform topology. This is a decisive step on the way to a rigorous proof of the
convergence of the normalized occupation measures towards an explicit mixture
of the maximizers, derived in [BKM15]. Our methods rely on deriving
H{\"o}lder-continuity of the Coulomb functional of the occupation measure with
exponentially small deviation probabilities and invoking the large deviation
theory developed in [MV14] to a certain shift-invariant functional of the
occupation measures.Comment: To appear in: "Annales de l'Institut Henri Poincare
Random walks conditioned to stay in Weyl chambers of type C and D
We construct the conditional versions of a multidimensional random walk given
that it does not leave the Weyl chambers of type C and of type D, respectively,
in terms of a Doob h-transform. Furthermore, we prove functional limit theorems
for the rescaled random walks. This is an extension of recent work by
Eichelsbacher and Koenig who studied the analogous conditioning for the Weyl
chamber of type A. Our proof follows recent work by Denisov and Wachtel who
used martingale properties and a strong approximation of random walks by
Brownian motion. Therefore, we are able to keep minimal moment assumptions.
Finally, we present an alternate function that is amenable to an h-transform in
the Weyl chamber of type C.Comment: 12 pages, submitted to EC
Large systems of path-repellent Brownian motions in a trap at positive temperature
We study a model of mutually repellent Brownian motions under
confinement to stay in some bounded region of space. Our model is defined in
terms of a transformed path measure under a trap Hamiltonian, which prevents
the motions from escaping to infinity, and a pair-interaction Hamiltonian,
which imposes a repellency of the paths. In fact, this interaction is an
-dependent regularisation of the Brownian intersection local times, an
object which is of independent interest in the theory of stochastic processes.
The time horizon (interpreted as the inverse temperature) is kept fixed. We
analyse the model for diverging number of Brownian motions in terms of a large
deviation principle. The resulting variational formula is the
positive-temperature analogue of the well-known Gross-Pitaevskii formula, which
approximates the ground state of a certain dilute large quantum system; the
kinetic energy term of that formula is replaced by a probabilistic energy
functional.
This study is a continuation of the analysis in \cite{ABK04} where we
considered the limit of diverging time (i.e., the zero-temperature limit) with
fixed number of Brownian motions, followed by the limit for diverging number of
motions.
\bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~K\"onig},
\newblock Large deviations for trapped interacting Brownian particles and
paths, \newblock {\it Ann. Probab.}, to appear (2004)
Associations between blood glucose and carotid intima-media thickness disappear after adjustment for shared risk factors: the KORA F4 study.
The association between blood glucose and carotid intima-media thickness (CIMT) is considered to be established knowledge. We aimed to assess whether associations between different measures of glycaemia and CIMT are actually independent of anthropometric variables and metabolic risk factors. Moreover, we checked published studies for the adjustment for shared risk factors of blood glucose and CIMT. Fasting glucose, 2-hour glucose, HbA1c, and CIMT were measured in 31-81-years-old participants of the population-based Cooperative Health Research in the Region of Augsburg (KORA) F4 study in Southern Germany (n = 2,663). CIMT was assessed according to the Rotterdam protocol. Linear and logistic regression models with adjustment for age, sex, anthropometric measures, hypertension, and dyslipidaemia were fitted to assess the association between continuous measures of glycaemia, and categories of glucose regulation, respectively, with CIMT. We found a 0.10 mm increase (95%-confidence interval: 0.08-0.12) in CIMT in subjects with compared to subjects without diabetes in crude analysis. This increase was not significant in age-sex adjusted models (p = 0.17). Likewise, neither impaired fasting glucose (p = 0.22) nor impaired glucose tolerance (p = 0.93) were associated with CIMT after adjustment for age, sex, and waist circumference. In multivariable adjusted models, age, sex, hypertension, waist circumference, HDL and LDL cholesterol, but neither fasting glucose nor 2-hour glucose nor HbA1c were associated with elevated CIMT. Literature findings are inconclusive regarding an independent association of glucose levels and CIMT. CIMT is highly dependent on traditional cardiovascular risk factors, but no relationships between blood glucose and CIMT were found after adjustment for age, sex, and anthropometric variables
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