28 research outputs found
Scaled limit and rate of convergence for the largest eigenvalue from the generalized Cauchy random matrix ensemble
In this paper, we are interested in the asymptotic properties for the largest
eigenvalue of the Hermitian random matrix ensemble, called the Generalized
Cauchy ensemble , whose eigenvalues PDF is given by
where is a complex number such
that and where is the size of the matrix ensemble. Using
results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that
for this ensemble, the largest eigenvalue divided by converges in law to
some probability distribution for all such that . Using
results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of
the largest eigenvalue for fixed , we also express the limiting probability
distribution in terms of some non-linear second order differential equation.
Eventually, we show that the convergence of the probability distribution
function of the re-scaled largest eigenvalue to the limiting one is at least of
order .Comment: Minor changes in this version. Added references. To appear in Journal
of Statistical Physic
New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain
In this paper we show how an infinite system of coupled Toda-type nonlinear
differential equations derived by one of us can be used efficiently to
calculate the time-dependent pair-correlations in the Ising chain in a
transverse field. The results are seen to match extremely well long large-time
asymptotic expansions newly derived here. For our initial conditions we use new
long asymptotic expansions for the equal-time pair correlation functions of the
transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising
model. Using this one can also study the equal-time wavevector-dependent
correlation function of the quantum chain, a.k.a. the q-dependent diagonal
susceptibility in the 2d Ising model, in great detail with very little
computational effort.Comment: LaTeX 2e, 31 pages, 8 figures (16 eps files). vs2: Two references
added and minor changes of style. vs3: Corrections made and reference adde
The Ising Susceptibility Scaling Function
We have dramatically extended the zero field susceptibility series at both
high and low temperature of the Ising model on the triangular and honeycomb
lattices, and used these data and newly available further terms for the square
lattice to calculate a number of terms in the scaling function expansion around
both the ferromagnetic and, for the square and honeycomb lattices, the
antiferromagnetic critical point.Comment: PDFLaTeX, 50 pages, 5 figures, zip file with series coefficients and
background data in Maple format provided with the source files. Vs2: Added
dedication and made several minor additions and corrections. Vs3: Minor
corrections. Vs4: No change to eprint. Added essential square-lattice series
input data (used in the calculation) that were removed from University of
Melbourne's websit
New insights into the genetic etiology of Alzheimer's disease and related dementias
Characterization of the genetic landscape of Alzheimer's disease (AD) and related dementias (ADD) provides a unique opportunity for a better understanding of the associated pathophysiological processes. We performed a two-stage genome-wide association study totaling 111,326 clinically diagnosed/'proxy' AD cases and 677,663 controls. We found 75 risk loci, of which 42 were new at the time of analysis. Pathway enrichment analyses confirmed the involvement of amyloid/tau pathways and highlighted microglia implication. Gene prioritization in the new loci identified 31 genes that were suggestive of new genetically associated processes, including the tumor necrosis factor alpha pathway through the linear ubiquitin chain assembly complex. We also built a new genetic risk score associated with the risk of future AD/dementia or progression from mild cognitive impairment to AD/dementia. The improvement in prediction led to a 1.6- to 1.9-fold increase in AD risk from the lowest to the highest decile, in addition to effects of age and the APOE Δ4 allele
Clearance of Minimal Residual Disease (MRD) in patients with 17p-CLL after Allogeneic HCT: Results of the Non-interventional Study of EBMT & ERIC
Development and application of statistical models for medical scientific researc