arXiv subjects

Leonid Petrov

Publications and source records attributed to Leonid Petrov.

At least 91 records · Page 5Linked to original sources

arXiv2014

Spectral theory for the q-Boson particle system

We develop spectral theory for the generator of the q-Boson (stochastic) particle system. Our central result is a Plancherel type isomorphism theorem for this system. This theorem has various implications. It proves the completeness of the Bethe ansatz for the q-Boson generator and consequently enables us to solve the Kolmogorov forward and backward equations for general initial data. Owing to a Markov duality with q-TASEP, this leads to moment formulas which characterize the fixed time distribution of q-TASEP started from general initial conditions. The theorem also implies the biorthogonality of the left and right eigenfunctions. We consider limits of our q-Boson results to a discrete delta Bose gas considered previously by van Diejen, as well as to another discrete delta Bose gas that describes the evolution of moments of the semi-discrete stochastic heat equation (or equivalently, the O'Connell-Yor semi-discrete directed polymer partition function). A further limit takes us to the delta Bose gas which arises in studying moments of the stochastic heat equation / Kardar-Parisi-Zhang equation.

math-ph↗

arXiv2013

Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes

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 in arXiv:math-ph/0507007. 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).

math.PR↗

arXiv2013

Nearest neighbor Markov dynamics on Macdonald processes

Macdonald processes are certain probability measures on two-dimensional arrays of interlacing particles introduced by Borodin and Corwin (arXiv:1111.4408 [math.PR]). They are defined in terms of nonnegative specializations of the Macdonald symmetric functions and depend on two parameters (q,t), where 0<= q, t < 1. Our main result is a classification of continuous time, nearest neighbor Markov dynamics on the space of interlacing arrays that act nicely on Macdonald processes. The classification unites known examples of such dynamics and also yields many new ones. When t = 0, one dynamics leads to a new integrable interacting particle system on the one-dimensional lattice, which is a q-deformation of the PushTASEP (= long-range TASEP). When q = t, the Macdonald processes become the Schur processes of Okounkov and Reshetikhin (arXiv:math/0107056 [math.CO]). In this degeneration, we discover new Robinson--Schensted-type correspondences between words and pairs of Young tableaux that govern some of our dynamics.

math.PR↗

arXiv2013

Australia Telescope Compact Array observations of Fermi unassociated sources

We report results of the first phase of observations with the Australia Telescope Compact Array (ATCA) at 5 and 9 GHz of the fields around 411 gamma-ray sources with declinations < +10 deg detected by Fermi but marked as unassociated in the 2FGL catalogue. We have detected 424 sources with flux densities in a range of 2 mJy to 6 Jy that lie within the 99 per cent localisation uncertainty of 283 gamma-ray sources. Of these, 146 objects were detected in both the 5 and 9 GHz bands. We found 84 sources in our sample with a spectral index flatter than -0.5. The majority of detected sources are weaker than 100 mJy and for this reason were not found in previous surveys. Approximately 1/3 of our sample, 128 objects, have the probability of being associated by more than 10 times than the probability of being a background source found in the vicinity of a gamma-ray object by chance. We present the catalogue of positions of these sources, estimates of their flux densities and spectral indices where available.

astro-ph.CO↗

arXiv2013

The catalogue of positions of optically bright extragalactic radio sources OBRS-2

It is anticipated that future space-born missions, such as Gaia, will be able to determine in optical domain positions of more than 100,000 bright quasars with sub-mas accuracies that are comparable to very long baseline interferometry (VLBI) accuracies. Comparisons of coordinate systems from space-born missions and from VLBI will be very important, first for investigation of possible systematic errors, second for investigation of possible shift between centroids of radio and optical emissions in active galaxy nuclea. In order to make such a comparison more robust, a program of densification of the grid of radio sources detectable with both VLBI and Gaia was launched in 2006. In the second observing campaign a set of 290 objects from the list of 398 compact extragalactic radio sources with declinations greater -10 deg was observed with the VLBA+EVN in 2010-2011 with the primary goal of producing their images with milliarcsecond resolution. These sources are brighter than 18 magnitude at V band. In this paper coordinates of observed sources have been derived with milliarcsecond accuracies from analysis of these VLBI observations following the method of absolute astrometry and their images were produced. The catalogue of positions of 295 target sources and estimates of their correlated flux densities at 2.2 and 8.4 GHz is presented. The accuracies of source coordinates are in a range of 2 to 200 mas, with the median 3.2 mas.

astro-ph.IM↗

arXiv2012

Early science with Korean VLBI network: the QCAL-1 43GHz calibrator survey

This paper presents the catalog of correlated flux densities in three ranges of baseline projection lengths of 637 sources from a 43 GHz (Q-band) survey observed with the Korean VLBI Network. Of them, 623 sources have not been observed before at Q-band with VLBI. The goal of this work in the early science phase of the new VLBI array is twofold: to evaluate the performance of the new instrument that operates in a frequency range of 22-129 GHz and to build a list of objects that can be used as targets and as calibrators. We have observed the list of 799 target sources with declinations down to -40 degrees. Among them, 724 were observed before with VLBI at 22 GHz and had correlated flux densities greater than 200 mJy. The overall detection rate is 78%. The detection limit, defined as the minimum flux density for a source to be detected with 90% probability in a single observation, was in a range of 115-180 mJy depending on declination. However, some sources as weak as 70 mJy have been detected. Of 623 detected sources, 33 objects are detected for the first time in VLBI mode. We determined their coordinates with the median formal uncertainty 20 mas. The results of this work set the basis for future efforts to build the complete flux-limited sample of extragalactic sources at frequencies 22 GHz and higher at 3/4 of the celestial sphere.

astro-ph.IM↗

arXiv2012

The Boundary of the Gelfand-Tsetlin Graph: New Proof of Borodin-Olshanski's Formula, and its q-analogue

In the recent paper [arXiv:1109.1412], Borodin and Olshanski have presented a novel proof of the celebrated Edrei-Voiculescu theorem which describes the boundary of the Gelfand-Tsetlin graph as a region in an infinite-dimensional coordinate space. This graph encodes branching of irreducible characters of finite-dimensional unitary groups. Points of the boundary of the Gelfand-Tsetlin graph can be identified with finite indecomposable (= extreme) characters of the infinite-dimensional unitary group. An equivalent description identifies the boundary with the set of doubly infinite totally nonnegative sequences. A principal ingredient of Borodin-Olshanski's proof is a new explicit determinantal formula for the number of semi-standard Young tableaux of a given skew shape (or of Gelfand-Tsetlin schemes of trapezoidal shape). We present a simpler and more direct derivation of that formula using the Cauchy-Binet summation involving the inverse Vandermonde matrix. We also obtain a q-generalization of that formula, namely, a new explicit determinantal formula for arbitrary q-specializations of skew Schur polynomials. Its particular case is related to the q-Gelfand-Tsetlin graph and q-Toeplitz matrices introduced and studied by Gorin [arXiv:1011.1769].

math.CO↗

arXiv2011

The KCAL VERA 22 GHz calibrator survey

We observed at 22 GHz with the VLBI array VERA a sample of 1536 sources with correlated flux densities brighter than 200 mJy at 8 GHz. One half of target sources has been detected. The detection limit was around 200 mJy. We derived the correlated flux densities of 877 detected sources in three ranges of projected baseline lengths. The objective of these observations was to determine the suitability of given sources as phase calibrators for dual-beam and phase-referencing observations at high frequencies. Preliminary results indicate that the number of compact extragalactic sources at 22 GHz brighter than a given correlated flux density level is twice less than at 8 GHz.

astro-ph.CO↗

arXiv2011

sl(2) Operators and Markov Processes on Branching Graphs

We present a unified approach to various examples of Markov dynamics on partitions studied by Borodin, Olshanski, Fulman, and the author. Our technique generalizes the Kerov's operators first appeared in [Okounkov, arXiv:math/0002135], and also stems from the study of duality of graded graphs in [Fomin, 1994]. Our main object is a countable branching graph carrying an sl(2,C)-module of a special kind. Using this structure, we introduce distinguished probability measures on the floors of the graph, and define two related types of Markov dynamics associated with these measures. We study spectral properties of the dynamics, and our main result is the explicit description of eigenfunctions of the Markov generator of one of the processes. For the Young graph our approach reconstructs the z-measures on partitions and the associated dynamics studied by Borodin and Olshanski [arXiv:math-ph/0409075, arXiv:0706.1034]. The generator of the dynamics of [arXiv:math-ph/0409075] is diagonal in the basis of the Meixner symmetric functions introduced recently by Olshanski [arXiv:1009.2037, arXiv:1103.5848]. We give new proofs to some of the results of these two papers. Other graphs to which our technique is applicable include the Pascal triangle, the Kingman graph (with the two-parameter Poisson-Dirichlet measures), the Schur graph and the general Young graph with Jack edge multiplicities.

math.CO↗

arXiv2011

VLBA observations of a complete sample of 2MASS galaxies

We are using the VLBA at 8 GHz to observe a sample of 834 nearby 2MASS galaxies that are stronger than 100 mJy in the NVSS. The goals of the project are to detect (1) supermassive black holes significantly offset from the IR positions of the host galaxy bulges and (2) binary black holes.

astro-ph.CO↗

arXiv2011

Precise absolute astrometry from the VLBA imaging and polarimetry survey at 5 GHz

We present in this paper accurate positions of 857 sources derived from the astrometric analysis of 16 eleven-hour experiments from the Very Long Baseline Array imaging and polarimetry survey at 5 GHz (VIPS). Among observed sources, positions of 430 objects were not determined before at a milliarcsecond level of accuracy. For 95% of the sources the uncertainty of their positions range from 0.3 to 0.9 mas, with the median value of 0.5 mas. This estimate of accuracy is substantiated by the comparison of positions of 386 sources that were previously observed in astrometric programs simultaneously at 2.3/8.6 GHz. Surprisingly, the ionosphere contribution to group delay was adequately modeled with the use of the total electron contents maps derived from GPS observations and only marginally affected estimates of source coordinates.

astro-ph.CO↗

arXiv2011

The catalogue of positions of optically bright extragalactic radio sources OBRS-1

It is expected that the European Space Agency mission Gaia will make possible to determine coordinates in the optical domain of more than 500000 quasars. In 2006, a radio astrometry project was launched with the overall goal to make comparison of coordinate systems derived from future space-born astrometry instruments with the coordinate system constructed from analysis of the global very long baseline interferometry (VLBI) more robust. Investigation of their rotation, zonal errors, and the non-alignment of the radio and optical positions caused by both radio and optical structures are important for validation of both techniques. In order to support these studies, the densification of the list of compact extragalactic objects that are bright in both radio and optical ranges is desirable. A set of 105 objects from the list of 398 compact extragalactic radio sources with declination > -10 deg was observed with the VLBA+EVN with the primary goal of producing their images with milliarcsecond resolution. These sources are brighter than 18 magnitude at V band, and they were previously detected at the European VLBI network. In this paper coordinates of observed sources have been derived with milliarcsecond accuracies from analysis of these VLBI observations following the method of absolute astrometry. The catalogue of positions of 105 target sources is presented. The accuracies of sources coordinates are in the range of 0.3 to 7 mas, with the median 1.1 mas.

astro-ph.CO↗

arXiv2011

On Measures on Partitions Arising in Harmonic Analysis for Linear and Projective Characters of the Infinite Symmetric Group

The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.

math.CO↗

arXiv2011

First geodetic observations using new VLBI stations ASKAP-29 and WARK12M

We report the results of a successful 7 hour 1.4 GHz VLBI experiment using two new stations, ASKAP-29 located in Western Australia and WARK12M located on the North Island of New Zealand. This was the first geodetic VLBI observing session with the participation of these new stations. We have determined the positions of ASKAP-29 and WARK12M. Random errors on position estimates are 150-200 mm for the vertical component and 40-50 mm for the horizontal component. Systematic errors caused by the unmodeled ionosphere path delay may reach 1.3 m for the vertical component.

astro-ph.IM↗

arXiv2011

The EVN Galactic Plane Survey - EGaPS

I present a catalogue of positions and correlated flux densities of 109 compact extragalactic radio sources in the Galactic plane determined from analysis of a 48 hour VLBI experiment at 22 GHz with the European VLBI Network. The median position uncertainty is 9 mas. The correlated flux densities of detected sources are in the range of 20 to 300 mJy. In addition to target sources, nine water masers have been detected, two of them new. I derived position of masers with accuracies 30 to 200 mas and determined velocities of maser components and their correlated flux densities. The catalogue and supporting material is available at http://astrogeo.org/egaps.

astro-ph.CO↗

arXiv2011

The LBA Calibrator Survey of southern compact extragalactic radio sources - LCS1

We present a catalogue of positions and correlated flux densities of 410 flat-spectrum, compact extragalactic radio sources, previously detected in the AT20G survey. The catalogue spans the declination range -90deg, -40deg and was constructed from four 24 hour VLBI observing sessions with the Australian Long Baseline Array made at 8.3 GHz. The detection rate in these experiments is 97%. The median uncertainty of source positions is 2.6 mas, the median correlated flux density at baseline projections lengths longer than 1000 km is 0.14 Jy. The goal of this work is 1) to provide a pool of sources with positions known at the milliarcsecond level of accuracy that are needed for phase referencing observations, for geodetic VLBI, and for space navigation; 2) to extend the complete flux-limited sample of compact extragalactic sources to the southern hemisphere; and 3) to investigate parsec-scale properties of high-frequency selected sources from the AT20G survey. As a result of the campaign, the number of compact radio sources with declinations < -40deg detectable with VLBI with measured correlated flux densities and positions known with the milliarcsec level of accuracies increased by a factor of 3.5. The catalogue and supporting material is available at http://astrogeo.org/lcs1 .

astro-ph.CO↗

arXiv2011

Pfaffian Stochastic Dynamics of Strict Partitions

We study a family of continuous time Markov jump processes on strict partitions (partitions with distinct parts) preserving the distributions introduced by Borodin (1997) in connection with projective representations of the infinite symmetric group. The one-dimensional distributions of the processes (i.e., the Borodin's measures) have determinantal structure. We express the dynamical correlation functions of the processes in terms of certain Pfaffians and give explicit formulas for both the static and dynamical correlation kernels using the Gauss hypergeometric function. Moreover, we are able to express our correlation kernels (both static and dynamical) through those of the z-measures on partitions obtained previously by Borodin and Olshanski in a series of papers. The results about the fixed time case were announced in the author's note arXiv:1002.2714. A part of the present paper contains proofs of those results.

math.PR↗

arXiv2010

Random Strict Partitions and Determinantal Point Processes

In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice {1,2,...} or on the open half-line (0,+\infty). The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.

math.PR↗