1,490 research outputs found
Differential equations for the KPZ and periodic KPZ fixed points
The KPZ fixed point is a 2d random field, conjectured to be the universal
limiting fluctuation field for the height function of models in the KPZ
universality class. Similarly, the periodic KPZ fixed point is a conjectured
universal field for spatially periodic models. For both fields, their
multi-point distributions in the space-time domain have been computed recently.
We show that for the case of the narrow-wedge initial condition, these
multi-point distributions can be expressed in terms of so-called integrable
operators. We then consider a class of operators that include the ones arising
from the KPZ and the periodic KPZ fixed points, and find that they are related
to various matrix integrable differential equations such as coupled matrix mKdV
equations, coupled matrix NLS equations with complex time, and matrix KP-II
equations. When applied to the KPZ fixed points, our results extend previously
known differential equations for one-point distributions and equal-time,
multi-position distributions to multi-time, multi-position setup
Recurrence of cervical cancer and its resistance to progestin therapy in a mouse model
Studies using K14E6/K14E7 transgenic mice expressing E6 and E7 oncoprotein of human papillomavirus type 16 (HPV16) have demonstrated that estrogen (E2) is required for the genesis and growth of cervical cancer. Our prior study using the same mouse model has showed that progestin drug medroxyprogesterone acetate (MPA) promotes regression of primary cervical cancer. In the present study, we use the same transgenic mouse model to determine whether the cancer recurs after MPA therapy. Cervical cancer recurred even if MPA treatment was continued. Unlike primary cervical cancer, the cancer recurred even in the absence of exogenous E2 when MPA treatment was ceased. Furthermore, recurrent cervical cancer did not fully regress upon MPA treatment. Our results support that MPA fails to completely eliminate primary cervical cancer cells and that remaining cancer cells grow independent of exogenous E2 and are refractory to MP
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
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
Airy processes and variational problems
We review the Airy processes; their formulation and how they are conjectured
to govern the large time, large distance spatial fluctuations of one
dimensional random growth models. We also describe formulas which express the
probabilities that they lie below a given curve as Fredholm determinants of
certain boundary value operators, and the several applications of these
formulas to variational problems involving Airy processes that arise in
physical problems, as well as to their local behaviour.Comment: Minor corrections. 41 pages, 4 figures. To appear as chapter in "PASI
Proceedings: Topics in percolative and disordered systems
The 1+1-dimensional Kardar-Parisi-Zhang equation and its universality class
We explain the exact solution of the 1+1 dimensional Kardar-Parisi-Zhang
equation with sharp wedge initial conditions. Thereby it is confirmed that the
continuum model belongs to the KPZ universality class, not only as regards to
scaling exponents but also as regards to the full probability distribution of
the height in the long time limit.Comment: Proceedings StatPhys 2
The k-Point Random Matrix Kernels Obtained from One-Point Supermatrix Models
The k-point correlation functions of the Gaussian Random Matrix Ensembles are
certain determinants of functions which depend on only two arguments. They are
referred to as kernels, since they are the building blocks of all correlations.
We show that the kernels are obtained, for arbitrary level number, directly
from supermatrix models for one-point functions. More precisely, the generating
functions of the one-point functions are equivalent to the kernels. This is
surprising, because it implies that already the one-point generating function
holds essential information about the k-point correlations. This also
establishes a link to the averaged ratios of spectral determinants, i.e. of
characteristic polynomials
On the partial connection between random matrices and interacting particle systems
In the last decade there has been increasing interest in the fields of random
matrices, interacting particle systems, stochastic growth models, and the
connections between these areas. For instance, several objects appearing in the
limit of large matrices arise also in the long time limit for interacting
particles and growth models. Examples of these are the famous Tracy-Widom
distribution functions and the Airy_2 process. The link is however sometimes
fragile. For example, the connection between the eigenvalues in the Gaussian
Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to
one-point distribution, and the connection breaks down if we consider the joint
distributions. In this paper we first discuss known relations between random
matrices and the asymmetric exclusion process (and a 2+1 dimensional
extension). Then, we show that the correlation functions of the eigenvalues of
the matrix minors for beta=2 Dyson's Brownian motion have, when restricted to
increasing times and decreasing matrix dimensions, the same correlation kernel
as in the 2+1 dimensional interacting particle system under diffusion scaling
limit. Finally, we analyze the analogous question for a diffusion on (complex)
sample covariance matrices.Comment: 31 pages, LaTeX; Added a section concerning the Markov property on
space-like path
Evidence for geometry-dependent universal fluctuations of the Kardar-Parisi-Zhang interfaces in liquid-crystal turbulence
We provide a comprehensive report on scale-invariant fluctuations of growing
interfaces in liquid-crystal turbulence, for which we recently found evidence
that they belong to the Kardar-Parisi-Zhang (KPZ) universality class for 1+1
dimensions [Phys. Rev. Lett. 104, 230601 (2010); Sci. Rep. 1, 34 (2011)]. Here
we investigate both circular and flat interfaces and report their statistics in
detail. First we demonstrate that their fluctuations show not only the KPZ
scaling exponents but beyond: they asymptotically share even the precise forms
of the distribution function and the spatial correlation function in common
with solvable models of the KPZ class, demonstrating also an intimate relation
to random matrix theory. We then determine other statistical properties for
which no exact theoretical predictions were made, in particular the temporal
correlation function and the persistence probabilities. Experimental results on
finite-time effects and extreme-value statistics are also presented. Throughout
the paper, emphasis is put on how the universal statistical properties depend
on the global geometry of the interfaces, i.e., whether the interfaces are
circular or flat. We thereby corroborate the powerful yet geometry-dependent
universality of the KPZ class, which governs growing interfaces driven out of
equilibrium.Comment: 31 pages, 21 figures, 1 table; references updated (v2,v3); Fig.19
updated & minor changes in text (v3); final version (v4); J. Stat. Phys.
Online First (2012
Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case
We consider the sample covariance matrices of large data matrices which have
i.i.d. complex matrix entries and which are non-square in the sense that the
difference between the number of rows and the number of columns tends to
infinity. We show that the second-order correlation function of the
characteristic polynomial of the sample covariance matrix is asymptotically
given by the sine kernel in the bulk of the spectrum and by the Airy kernel at
the edge of the spectrum. Similar results are given for real sample covariance
matrices
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