2,126 research outputs found
Probability densities and distributions for spiked and general variance Wishart -ensembles
A Wishart matrix is said to be spiked when the underlying covariance matrix
has a single eigenvalue different from unity. As increases through
, a gap forms from the largest eigenvalue to the rest of the spectrum, and
with of order the scaled largest eigenvalues form a well
defined parameter dependent state. Recent works by Bloemendal and Vir\'ag [BV],
and Mo, have quantified this parameter dependent state for real Wishart
matrices from different viewpoints, and the former authors have done similarly
for the spiked Wishart -ensemble. The latter is defined in terms of
certain random bidiagonal matrices. We use a recursive structure to give an
alternative construction of the spiked and more generally the general variance
Wishart -ensemble, and we give the exact form of the joint eigenvalue
PDF for the two matrices in the recurrence. In the case of real quaternion
Wishart matrices () the latter is recognised as having appeared in
earlier studies on symmetrized last passage percolation, allowing the exact
form of the scaled distribution of the largest eigenvalue to be given. This
extends and simplifies earlier work of Wang, and is an alternative derivation
to a result in [BV]. We also use the construction of the spiked Wishart
-ensemble from [BV] to give a simple derivation of the explicit form of
the eigenvalue PDF.Comment: 18 page
Symmetrized models of last passage percolation and non-intersecting lattice paths
It has been shown that the last passage time in certain symmetrized models of
directed percolation can be written in terms of averages over random matrices
from the classical groups , and . We present a theory of
such results based on non-intersecting lattice paths, and integration
techniques familiar from the theory of random matrices. Detailed derivations of
probabilities relating to two further symmetrizations are also given.Comment: 21 pages, 5 figure
Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
The one-dimensional totally asymmetric simple exclusion process (TASEP) is
considered. We study the time evolution property of a tagged particle in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the dynamics of
TASEP to the Schur process, we show that the function is represented as the
Fredholm determinant. We also study the scaling limit. The universality of the
largest eigenvalue in the random matrix theory is realized in the limit. When
the hopping rates of all particles are the same, it is found that the joint
distribution function converges to that of the Airy process after the time at
which the particle begins to move. On the other hand, when there are several
particles with small hopping rate in front of a tagged particle, the limiting
process changes at a certain time from the Airy process to the process of the
largest eigenvalue in the Hermitian multi-matrix model with external sources.Comment: 48 pages, 8 figure
Polynuclear growth model, GOE and random matrix with deterministic source
We present a random matrix interpretation of the distribution functions which
have appeared in the study of the one-dimensional polynuclear growth (PNG)
model with external sources. It is shown that the distribution, GOE, which
is defined as the square of the GOE Tracy-Widom distribution, can be obtained
as the scaled largest eigenvalue distribution of a special case of a random
matrix model with a deterministic source, which have been studied in a
different context previously. Compared to the original interpretation of the
GOE as ``the square of GOE'', ours has an advantage that it can also
describe the transition from the GUE Tracy-Widom distribution to the GOE.
We further demonstrate that our random matrix interpretation can be obtained
naturally by noting the similarity of the topology between a certain
non-colliding Brownian motion model and the multi-layer PNG model with an
external source. This provides us with a multi-matrix model interpretation of
the multi-point height distributions of the PNG model with an external source.Comment: 27pages, 4 figure
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Distinct mechanisms of Drosophila CRYPTOCHROME-mediated light-evoked membrane depolarization and in vivo clock resetting.
Drosophila CRYPTOCHROME (dCRY) mediates electrophysiological depolarization and circadian clock resetting in response to blue or ultraviolet (UV) light. These light-evoked biological responses operate at different timescales and possibly through different mechanisms. Whether electron transfer down a conserved chain of tryptophan residues underlies biological responses following dCRY light activation has been controversial. To examine these issues in in vivo and in ex vivo whole-brain preparations, we generated transgenic flies expressing tryptophan mutant dCRYs in the conserved electron transfer chain and then measured neuronal electrophysiological phototransduction and behavioral responses to light. Electrophysiological-evoked potential analysis shows that dCRY mediates UV and blue-light-evoked depolarizations that are long lasting, persisting for nearly a minute. Surprisingly, dCRY appears to mediate red-light-evoked depolarization in wild-type flies, absent in both cry-null flies, and following acute treatment with the flavin-specific inhibitor diphenyleneiodonium in wild-type flies. This suggests a previously unsuspected functional signaling role for a neutral semiquinone flavin state (FADH•) for dCRY. The W420 tryptophan residue located closest to the FAD-dCRY interaction site is critical for blue- and UV-light-evoked electrophysiological responses, while other tryptophan residues within electron transfer distance to W420 do not appear to be required for light-evoked electrophysiological responses. Mutation of the dCRY tryptophan residue W342, more distant from the FAD interaction site, mimics the cry-null behavioral light response to constant light exposure. These data indicate that light-evoked dCRY electrical depolarization and clock resetting are mediated by distinct mechanisms
Echo statistics associated with discrete scatterers: A tutorial on physics-based methods
Author Posting. © Acoustical Society of America, 2018. This article is posted here by permission of Acoustical Society of America for personal use, not for redistribution. The definitive version was published in Journal of the Acoustical Society of America, 144(6), (2018): 3124-3171. doi: 10.1121/1.5052255.When a beam emitted from an active monostatic sensor system sweeps across a volume, the echoes from scatterers present will fluctuate from ping to ping due to various interference phenomena and statistical processes. Observations of these fluctuations can be used, in combination with models, to infer properties of the scatterers such as numerical density. Modeling the fluctuations can also help predict system performance and associated uncertainties in expected echoes. This tutorial focuses on “physics-based statistics,” which is a predictive form of modeling the fluctuations. The modeling is based principally on the physics of the scattering by individual scatterers, addition of echoes from randomized multiple scatterers, system effects involving the beampattern and signal type, and signal theory including matched filter processing. Some consideration is also given to environment-specific effects such as the presence of boundaries and heterogeneities in the medium. Although the modeling was inspired by applications of sonar in the field of underwater acoustics, the material is presented in a general form, and involving only scalar fields. Therefore, it is broadly applicable to other areas such as medical ultrasound, non-destructive acoustic testing, in-air acoustics, as well as radar and lasers.The content of this work is based on research conducted in the past from years of support from the U.S. Office of Naval Research and the Woods Hole Oceanographic Institution, Woods Hole, MA. Writing of the manuscript by W.-J.L. was also supported by the Science and Engineering Enrichment and Development Postdoctoral Fellowship from the Applied Physics Laboratory, University of Washington, WA. The authors are grateful to Dr. Benjamin A. Jones of the Naval Postgraduate School, Monterey, CA for his thoughtful suggestions on an early draft of the manuscript. The authors are also grateful to the reviewer for the in-depth and constructive recommendations. W.-J.L. and K.B. contributed equally to this work.2019-06-0
Random walks and random fixed-point free involutions
A bijection is given between fixed point free involutions of
with maximum decreasing subsequence size and two classes of vicious
(non-intersecting) random walker configurations confined to the half line
lattice points . In one class of walker configurations the maximum
displacement of the right most walker is . Because the scaled distribution
of the maximum decreasing subsequence size is known to be in the soft edge GOE
(random real symmetric matrices) universality class, the same holds true for
the scaled distribution of the maximum displacement of the right most walker.Comment: 10 page
Finite time corrections in KPZ growth models
We consider some models in the Kardar-Parisi-Zhang universality class, namely
the polynuclear growth model and the totally/partially asymmetric simple
exclusion process. For these models, in the limit of large time t, universality
of fluctuations has been previously obtained. In this paper we consider the
convergence to the limiting distributions and determine the (non-universal)
first order corrections, which turn out to be a non-random shift of order
t^{-1/3} (of order 1 in microscopic units). Subtracting this deterministic
correction, the convergence is then of order t^{-2/3}. We also determine the
strength of asymmetry in the exclusion process for which the shift is zero.
Finally, we discuss to what extend the discreteness of the model has an effect
on the fitting functions.Comment: 34 pages, 5 figures, LaTeX; Improved version including shift of PASEP
height functio
Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes
Random Hermitian matrices with a source term arise, for instance, in the
study of non-intersecting Brownian walkers \cite{Adler:2009a, Daems:2007} and
sample covariance matrices \cite{Baik:2005}.
We consider the case when the external source matrix has two
distinct real eigenvalues: with multiplicity and zero with multiplicity
. The source is small in the sense that is finite or , for . For a Gaussian potential, P\'ech\'e
\cite{Peche:2006} showed that for sufficiently small (the subcritical
regime) the external source has no leading-order effect on the eigenvalues,
while for sufficiently large (the supercritical regime) eigenvalues
exit the bulk of the spectrum and behave as the eigenvalues of
Gaussian unitary ensemble (GUE). We establish the universality of these results
for a general class of analytic potentials in the supercritical and subcritical
regimes.Comment: 41 pages, 4 figure
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