Abstract The very long baseline interferometry technique allows us to determine the positions of thousands of radio sources using the absolute astrometry approach. I have investigated the impacts of ...a selection of observing frequencies in a range from 2 to 43 GHz in single-band, dual-band, and quad-band observing modes on astrometric results. I processed seven data sets in a range of 72,000 to 6.9 million observations, estimated source positions, and compared them. I found that source positions derived from dual-band, quad-band, and 23.6 GHz single-band data agree at a level below 0.2 mas. Comparison of independent data sets allowed me to assess the error levels of individual catalogs: 0.05–0.07 mas per position component. Further comparison showed that individual catalogs have systematic errors at the same level. The positions from 23.6 GHz single-band data show systematic errors related to the residual ionosphere contribution. Analysis of source position differences revealed systematic errors along jet directions at a level of 0.09 mas. Network-related systematic errors affect all the data, regardless of frequency. Comparison of position estimates allowed me to derive the stochastic error model that closes the error budget. Based on the collected evidence, I have made a conclusion that the development of frequency-dependent reference frames of the entire sky is not warranted. In most cases dual-band, quad-band, and single-band data at a frequency of 22 GHz and higher can be used interchangeably, which allows us to exploit the strength of a specific frequency setup for given objects. Mixing observations at different frequencies causes errors not exceeding 0.07 mas.
The ionospheric path delay impacts single-band very long baseline interferometry (VLBI) group delays, which limits their applicability for absolute astrometry. I consider two important cases: when ...observations are made simultaneously in two bands, but delays in only one band are available for a subset of observations and when observations are made at one band design. I developed optimal procedures of data analysis for both cases using Global Navigation Satellite System (GNSS) ionosphere maps, provided a stochastic model that describes ionospheric errors, and evaluated their impact on source position estimates. I demonstrate that the stochastic model is accurate at a level of 15%. I found that using GNSS ionospheric maps as is introduces serious biases in estimates of declination and I developed a procedure that almost eliminates them. I found serendipitously that GNSS ionospheric maps have multiplicative errors and have to be scaled by 0.85 in order to mitigate the declination bias. A similar scale factor was found in comparison of the vertical total electron content from satellite altimetry against GNSS ionospheric maps. I favor interpretation of this scaling factor as a manifestation of the inadequacy of the thin shell model of the ionosphere. I showed that we are able to model the ionospheric path delay to the extent that no noticeable systematic errors emerge and we are able to assess adequately the contribution of the ionosphere-driven random errors on source positions. This makes single-band absolute astrometry a viable option that can be used for source position determination.
A Gelfand–Tsetlin scheme of depth
N
is a triangular array with
m
integers at level
m
,
m
=
1
,
…
,
N
, subject to certain interlacing constraints. We study the ensemble of uniformly random ...Gelfand–Tsetlin schemes with arbitrary fixed
N
th row. We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its
q
-deformation). This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302,
2007
). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).
We introduce a four-parameter family of interacting particle systems on the line, which can be diagonalized explicitly via a complete set of Bethe ansatz eigenfunctions, and which enjoy certain ...Markov dualities. Using this, for the systems started in step initial data, we write down nested contour integral formulas for moments and Fredholm determinant formulas for Laplace-type transforms. Taking various choices or limits of parameters, this family degenerates to many of the known exactly solvable models in the Kardar–Parisi–Zhang universality class, as well as leads to many new examples of such models. In particular, asymmetric simple exclusion process, the stochastic six-vertex model,
q
-totally asymmetric simple exclusion process and various directed polymer models all arise in this manner. Our systems are constructed from stochastic versions of the
R
-matrix related to the six-vertex model. One of the key tools used here is the fusion of
R
-matrices and we provide a probabilistic proof of this procedure.
We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point
q
-moments of the height function ...and for the
q
-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1
+
1)d KPZ universality class, including the stochastic six vertex model, ASEP, various
q
-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in Borodin (On a family of symmetric rational functions,
2014
); they also generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six vertex model.
Spin
q
-Whittaker symmetric polynomials labeled by partitions
λ
were recently introduced by Borodin and Wheeler (Spin
q
-Whittaker Polynomials, 2017. arXiv preprint
arXiv:1701.06292
math.CO) in the ...context of integrable
sl
2
vertex models. They are a one-parameter deformation of the
t
=
0
Macdonald polynomials. We present a new more convenient modification of spin
q
-Whittaker polynomials and find two Macdonald type
q
-difference operators acting diagonally in these polynomials with eigenvalues, respectively,
q
-
λ
1
and
q
λ
N
(where
λ
is the polynomial’s label). We study probability measures on interlacing arrays based on spin
q
-Whittaker polynomials, and match their observables with known stochastic particle systems such as the
q
-Hahn TASEP. In a scaling limit as
q
↗
1
, spin
q
-Whittaker polynomials turn into a new one-parameter deformation of the
gl
n
Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as
q
↗
1
we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016.
arXiv:1503.04117
math.PR), and relate it to spin Whittaker functions.
We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common ...properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are
not
determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the
4
×
4
problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size
n
≥
4
, which appear new for
n
≥
5
. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.
This paper presents the results of the largest very long baseline interferometry (VLBI) absolute astrometry campaign to date of 13,645 radio source observations with the Very Long Baseline Array. Of ...these, 7220 have been detected, including 6755 target sources that have never been observed with VLBI before. This makes the present VLBI catalog the largest ever published. The positions of the target sources have been determined with the median uncertainty of 1.7 mas, and 15,542 images of 7171 sources have been generated. Unlike previous absolute radio astrometry campaigns, observations were made at 4.3 and 7.6 GHz simultaneously using a single wide-band receiver. Because of the fine spectral and time resolutions, the field of view was 4'–8'—much greater than the 10''–20'' in previous surveys. This made possible the use of input catalogs with low position accuracy and the detection of a compact component in extended sources. Unlike previous absolute astrometry campaigns, both steep- and flat-spectrum sources were observed. The observations were scheduled in the so-called filler mode to fill the gaps between other high-priority programs. This was achieved by the development of the totally automatic scheduling procedure.
Mapping TASEP back in time Petrov, Leonid; Saenz, Axel
Probability theory and related fields,
02/2022, Letnik:
182, Številka:
1-2
Journal Article
Recenzirano
Odprti dostop
We obtain a new relation between the distributions
μ
t
at different times
t
≥
0
of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. ...Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions
μ
t
backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving
μ
t
which in turn brings new identities for expectations with respect to
μ
t
. The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.
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