Borodin, Alexei
Overview
Works:  36 works in 38 publications in 1 language and 51 library holdings 

Roles:  Author, Thesis advisor 
Publication Timeline
.
Most widely held works by
Alexei Borodin
Large time asymptotics of growth models on spacelike paths II: PNG and parallel TASEP by
Alexei Borodin(
Book
)
2 editions published in 2007 in English and held by 3 WorldCat member libraries worldwide
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any spacelike path in the spacetime plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update
2 editions published in 2007 in English and held by 3 WorldCat member libraries worldwide
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any spacelike path in the spacetime plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update
Anisotropic growth of random surfaces in 2+1 dimensions by
Alexei Borodin(
Book
)
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2008 in English and held by 3 WorldCat member libraries worldwide
Transition between Airy1 and Airy2 processes and TASEP fluctuations by
Alexei Borodin(
Book
)
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2007 in English and held by 3 WorldCat member libraries worldwide
Large time asymptotics of growth models on spacelike paths I: PushASEP by
Alexei Borodin(
Book
)
2 editions published in 2007 in English and held by 3 WorldCat member libraries worldwide
We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases
2 editions published in 2007 in English and held by 3 WorldCat member libraries worldwide
We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases
Free energy fluctuations for directed polymers in random media in 1 + 1 dimensions by
Alexei Borodin(
Book
)
1 edition published in 2012 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2012 in English and held by 3 WorldCat member libraries worldwide
Harmonic analysis on the infinite symmetric group by
Alexei Borodin(
)
1 edition published in 2001 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2001 in English and held by 2 WorldCat member libraries worldwide
Large time asymptotics of growth models on spacelike paths II: PNG and parallel TASEP(
)
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any spacelike path in the spacetime plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider the polynuclear growth (PNG) model in 1+1 dimension with flat initial condition and no extra constraints. The joint distributions of surface height at finitely many points at a fixed time moment are given as marginals of a signed determinantal point process. The long time scaling limit of the surface height is shown to coincide with the Airy$_1$ process. This result holds more generally for the observation points located along any spacelike path in the spacetime plane. We also obtain the corresponding results for the discrete time TASEP (totally asymmetric simple exclusion process) with parallel update
Anisotropic growth of random surfaces in 2 + 1 dimensions(
)
1 edition published in 2008 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2008 in English and held by 2 WorldCat member libraries worldwide
Transition between Airy1 and Airy2 processes and TASEP fluctuations(
)
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. F or large time $t$, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy$_1$ and Airy$_2$ processes, whose onepoint distributions are the GOE and GUE TracyWidom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its onepoint distribution is a new interpolati on between GOE and GUE edge distributions
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider the totally asymmetric simple exclusion process, a model in the KPZ universality class. We focus on the fluctuations of particle positions starting with certain deterministic initial conditions. F or large time $t$, one has regions with constant and linearly decreasing density. The fluctuations on these two regions are given by the Airy$_1$ and Airy$_2$ processes, whose onepoint distributions are the GOE and GUE TracyWidom distributions of random matrix theory. In this paper we analyze the transition region between these two regimes and obtain the transition process. Its onepoint distribution is a new interpolati on between GOE and GUE edge distributions
Large time asymptotics of growth models on spacelike paths I: PushASEP(
)
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases
1 edition published in 2007 in English and held by 2 WorldCat member libraries worldwide
We consider a new interacting particle system on the onedimensional lattice that interpolates between TASEP and Toom's model: A particle cannot jump to the right if the neighboring site is occupied, and when jumping to the left it simply pushes all the neighbors that block its way. We prove that for flat and step initial conditions, the large time fluctuations of the height function of the associated growth model along any spacelike path are described by the Airy$_1$ and Airy$_2$ processes. This includes fluctuations of the height profile for a fixed time and fluctuations of a tagged particle's trajectory as special cases
Infinitedimensional diffusions as limits of random walks on partitions by
Alexei Borodin(
)
1 edition published in 2009 in Undetermined and held by 1 WorldCat member library worldwide
Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinitedimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the socalled zmeasures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described
1 edition published in 2009 in Undetermined and held by 1 WorldCat member library worldwide
Starting with finite Markov chains on partitions of a natural number n we construct, via a scaling limit transition as n → ∞, a family of infinitedimensional diffusion processes. The limit processes are ergodic; their stationary distributions, the socalled zmeasures, appeared earlier in the problem of harmonic analysis for the infinite symmetric group. The generators of the processes are explicitly described
Markov processes on the path space of the Gelfand–Tsetlin graph and on its boundary by
Alexei Borodin(
)
1 edition published in 2012 in Undetermined and held by 1 WorldCat member library worldwide
We construct a fourparameter family of Markov processes on infinite Gelfand–Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinitedimensional boundary of the Gelfand–Tsetlin graph or, equivalently, the space of extreme characters of the infinitedimensional unitary group U(∞). The process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on U(∞) posed in Olshanski (2003) [44]. As was shown in Borodin and Olshanski (2005) [11], this measure can also be described as a determinantal point process with a correlation kernel expressed through the Gauss hypergeometric function
1 edition published in 2012 in Undetermined and held by 1 WorldCat member library worldwide
We construct a fourparameter family of Markov processes on infinite Gelfand–Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinitedimensional boundary of the Gelfand–Tsetlin graph or, equivalently, the space of extreme characters of the infinitedimensional unitary group U(∞). The process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on U(∞) posed in Olshanski (2003) [44]. As was shown in Borodin and Olshanski (2005) [11], this measure can also be described as a determinantal point process with a correlation kernel expressed through the Gauss hypergeometric function
On adding a list of numbers (and other onedependent determinantal processes) by
Alexei Borodin(
)
1 edition published in 2010 in Undetermined and held by 1 WorldCat member library worldwide
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a onedependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae, and that any onedependent point process on the integers is determinantal. The examples give a gentle introduction to the emerging fields of onedependent and determinantal point processes
1 edition published in 2010 in Undetermined and held by 1 WorldCat member library worldwide
Adding a column of numbers produces "carries" along the way. We show that random digits produce a pattern of carries with a neat probabilistic description: the carries form a onedependent determinantal point process. This makes it easy to answer natural questions: How many carries are typical? Where are they located? We show that many further examples, from combinatorics, algebra and group theory, have essentially the same neat formulae, and that any onedependent point process on the integers is determinantal. The examples give a gentle introduction to the emerging fields of onedependent and determinantal point processes
Limits of detenninantal processes near a tacnode by
Alexei Borodin(
)
1 edition published in 2011 in Undetermined and held by 1 WorldCat member library worldwide
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process
1 edition published in 2011 in Undetermined and held by 1 WorldCat member library worldwide
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process
Gibbs Ensembles of Nonintersecting Paths by
Alexei Borodin(
)
1 edition published in 2010 in Undetermined and held by 1 WorldCat member library worldwide
We consider a family of determinantal random point processes on the twodimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are nontranslationinvariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices
1 edition published in 2010 in Undetermined and held by 1 WorldCat member library worldwide
We consider a family of determinantal random point processes on the twodimensional lattice and prove that members of our family can be interpreted as a kind of Gibbs ensembles of nonintersecting paths. Examples include probability measures on lozenge and domino tilings of the plane, some of which are nontranslationinvariant. The correlation kernels of our processes can be viewed as extensions of the discrete sine kernel, and we show that the Gibbs property is a consequence of simple linear relations satisfied by these kernels. The processes depend on infinitely many parameters, which are closely related to parametrization of totally positive Toeplitz matrices
On a conjecture of Widom by
Alexei Borodin(
)
1 edition published in 2006 in Undetermined and held by 1 WorldCat member library worldwide
We prove a conjecture of Widom (2002 Int. Math. Res. Not. 455–64 (Preprint math/0108008)) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct, we obtain a new proof of Okounkov's formula for the (determinantal) correlation functions of the Schur measures on partitions
1 edition published in 2006 in Undetermined and held by 1 WorldCat member library worldwide
We prove a conjecture of Widom (2002 Int. Math. Res. Not. 455–64 (Preprint math/0108008)) about the reality of eigenvalues of certain infinite matrices arising in asymptotic analysis of large Toeplitz determinants. As a byproduct, we obtain a new proof of Okounkov's formula for the (determinantal) correlation functions of the Schur measures on partitions
Harmonic Functions on Multiplicative Graphs and Interpolation Polynomials by
Alexei Borodin(
)
1 edition published in 2000 in Undetermined and held by 1 WorldCat member library worldwide
We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a byproduct, we compute certain Selbergtype integrals
1 edition published in 2000 in Undetermined and held by 1 WorldCat member library worldwide
We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur's S and P functions and with Jack symmetric functions. As a byproduct, we compute certain Selbergtype integrals
Limits of determinantal processes near a tacnode by
Alexei Borodin(
)
1 edition published in 2011 in Undetermined and held by 1 WorldCat member library worldwide
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process
1 edition published in 2011 in Undetermined and held by 1 WorldCat member library worldwide
We study a Markov process on a system of interlacing particles. At large times the particles fill a domain that depends on a parameter ε > 0. The domain has two cusps, one pointing up and one pointing down. In the limit ε ↓ 0 the cusps touch, thus forming a tacnode. The main result of the paper is a derivation of the local correlation kernel around the tacnode in the transition regime ε ↓ 0. We also prove that the local process interpolates between the Pearcey process and the GUE minor process
A note on the Pfaffian integration theorem by
Alexei Borodin(
)
1 edition published in 2007 in Undetermined and held by 1 WorldCat member library worldwide
Two alternative, fairly compact proofs are presented of the Pfaffian integration theorem that surfaced in the recent studies of spectral properties of Ginibre's Orthogonal Ensemble. The first proof is based on a concept of the Fredholm Pfaffian; the second proof is purely linear algebraic
1 edition published in 2007 in Undetermined and held by 1 WorldCat member library worldwide
Two alternative, fairly compact proofs are presented of the Pfaffian integration theorem that surfaced in the recent studies of spectral properties of Ginibre's Orthogonal Ensemble. The first proof is based on a concept of the Fredholm Pfaffian; the second proof is purely linear algebraic
Asymptotics of Plancherel measures for the infinitedimensional unitary group by
Alexei Borodin(
)
1 edition published in 2008 in Undetermined and held by 1 WorldCat member library worldwide
We study a twodimensional family of probability measures on infinite GelfandTsetlin schemes induced by a distinguished family of extreme characters of the infinitedimensional unitary group. These measures are unitary group analogs of the wellknown Plancherel measures for symmetric groups. We show that any measure from our family defines a determinantal point process on Z_+ x Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes
1 edition published in 2008 in Undetermined and held by 1 WorldCat member library worldwide
We study a twodimensional family of probability measures on infinite GelfandTsetlin schemes induced by a distinguished family of extreme characters of the infinitedimensional unitary group. These measures are unitary group analogs of the wellknown Plancherel measures for symmetric groups. We show that any measure from our family defines a determinantal point process on Z_+ x Z, and we prove that in appropriate scaling limits, such processes converge to two different extensions of the discrete sine process as well as to the extended Airy and Pearcey processes
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Alexei Borodin mathmaticien russe
Alexei Borodin RussianAmerican mathematician
Бородин, Алексей Михайлович
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