Noncommutative geometry of random surfaces (2009)
We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a...
The quantum differential equation of the Hilbert scheme of points in the plane (2009)
Okounkov, Andrei, Pandharipande, Rahul
We discuss here basic properties of the quantum differential equation of the Hilbert scheme of points in the plane. Our emphasis is on intertwining operators (which shift equivariant parameters) and...
Exts and Vertex Operators (2008)
Carlsson, Erik, Okounkov, Andrei
The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We...
Abstract. We consider 3-parametric polynomials P (x; q; t; s) which replace the An-series interpolation Macdonald polynomials P (x; q; t) for the BCn-type root system. For these polynomials we prove...
BINOMIAL FORMULA FOR CHARACTERS OF CLASSICAL GROUPS AND APPLICATIONS (2007)
Andrei Okounkov, Grigori Olshanski
Abstract. Let G be any of the complex classical groups GL(n) , SO(2n + 1), Sp(2n) , O(2n) , let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal...
ON NEWTON INTERPOLATION OF SYMMETRIC FUNCTIONS. A CHARACTERIZATION OF (2007)
Interpolation Macdonald Polynomials, Andrei Okounkov
1. Brief introduction 2. General interpolation problem 3. Examples of interpolation polynomials
ON THE REPRESENTATIONS OF THE INFINITE SYMMETRIC GROUP (2007)
Abstract. We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group S(1). In particular, our methods yield two simple proofs of the...
Submitted to Transformation Groups QUANTUM IMMANANTS AND HIGHER CAPELLI IDENTITIES (2007)
Abstract. We consider remarkable central elements of the universal enveloping algebra U(gl(n)) which we call quantum immanants. We express them in terms of generators E ij of U(gl(n)) and as...
ON NEWTON INTERPOLATION OF SYMMETRIC FUNCTIONS. A CHARACTERIZATION OF INTERPOLATION (2007)
Macdonald Polynomials, Andrei Okounkov
2. General interpolation problem 3. Examples of interpolation polynomials
Abstract. We generalize the binomial formula for Jack polynomials proved in [OO2] and consider some applications. x1 Binomial formula Binomial type theorems (that is Taylor and Newton interpolation...
(x; q; t) for other classical root systems. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy...
THE BOUNDARY OF YOUNG GRAPH (2007)
Edge Multiplicities, Sergei Kerov, Andrei Okounkov, Grigori Olshanski
Abstract. Consider the lattice of all Young diagrams ordered by inclusion, and denote by Yits Hasse graph. Using the Pieri formula for Jack symmetric polynomials, we endow the edges of the graph...
Geometry of planar log-fronts (2006)
Mikhalkin, Grigory, Okounkov, Andrei
The log-front of two plane curves P(z,w)=0 and Q(z,w)=0 is the locus of (a,b) such that Q(az,bw)=0 is tangent to P(z,w)=0. Log-fronts generalize dual curves, wave fronts, and arise naturally in the...
Limits of BC-type orthogonal polynomials as the number of variables goes to infinity (2006)
Okounkov, Andrei, Olshanski, Grigori
We describe the asymptotic behavior of the multivariate BC-type Jacobi polynomials as the number of variables and the Young diagram indexing the polynomial go to infinity. In particular, our results...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph $G$ embedded in the plane. We derive...
Planar dimers and Harnack curves (2006)
Kenyon, Richard, Okounkov, Andrei
In this article we study the connection between dimers and Harnack curves discovered in [15]. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the...
Random partitions and instanton counting (2006)
In this expository paper we summarize the connection between random partitions and 4-dimensional supersymmetric gauge theories discovered in hep-th/0306238. There is also a brief section about how...
Kenyon, Richard, Sheffield, Scott, Okounkov, Andrei
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive...
Limit shapes and the complex burgers equation (2005)
Kenyon, Richard, Okounkov, Andrei
In this paper we study surfaces in R^3 that arise as limit shapes in a class of random surface models arising from dimer models. The limit shapes are minimizers of a surface tension functional, that...
Pillowcases and quasimodular forms (2005)
We prove that natural generating functions for enumeration of branched coverings of the pillowcase orbifold are level 2 quasimodular forms. This gives a way to compute the volumes of the strata of...
Random skew plane partitions and the Pearcey process (2005)
Okounkov, Andrei, Reshetikhin, Nicolai
We study random skew 3D partitions weighted by $q^{\textup{vol}}$ and, specifically, the $q\to 1$ asymptotics of local correlations near various points of the limit shape. We obtain sine-kernel...
Random surfaces enumerating algebraic curves (2004)
These are notes from my lecture at 4ECM in Stockholm (June 2004).
The theta characteristic of a branched covering (2003)
Eskin, Alex, Okounkov, Andrei, Pandharipande, Rahul
We give a group-theoretic description of the parity of a pull-back of a theta characteristic under a branched covering. It involves lifting monodromy of the covering to the semidirect product of the...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on an weighted, bipartite, doubly periodic graph G embedded in the plane. We derive...
Planar dimers and Harnack curves (2003)
Kenyon, Richard, Okounkov, Andrei
In this paper we study the connection between dimers and Harnack curves discovered in math-ph/0311005. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove...
Quantum Calabi-Yau and Classical Crystals (2003)
Okounkov, Andrei, Reshetikhin, Nikolai, Vafa, Cumrun
We propose a new duality involving topological strings in the limit of large string coupling constant. The dual is described in terms of a classical statistical mechanical model of crystal melting,...
The uses of random partitions (2003)
These are extended notes for my talk at the ICMP 2003 in Lisbon. Our goal here is to demonstrate how natural and fundamental random partitions are from many different points of view. We discuss...
Symmetric functions and random partitions (2003)
These are (not updated) notes from the lectures I gave at the NATO ASI ``Symmetric Functions 2001'' at the Isaac Newton Institute in Cambridge (June 25 -- July 6, 2001). Their goal is an informal...
Random trees and moduli of curves (2003)
These are (not updated) notes from the lectures I gave in St.Petersburg in July of 2001. Their goal is to give an expository account of the proof of Kontsevich's combinatorial formula for...
Virasoro constraints for target curves (2003)
Okounkov, Andrei, Pandharipande, Rahul
We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization,...
Seiberg-Witten Theory and Random Partitions (2003)
Nekrasov, Nikita, Okounkov, Andrei
We study N=2 supersymmetric four dimensional gauge theories, in a certain N=2 supergravity background, called Omega-background. The partition function of the theory in the Omega-background can be...
The equivariant Gromov-Witten theory of P^1 (2002)
Okounkov, Andrei, Pandharipande, Rahul
We express all equivariant Gromov-Witten invariants of the projective line as matrix elements of explicit operators acting in the Fock space. As a consequence, we prove the equivariant theory is...
Multipoint series of Gromov-Witten invariants of CP^1 (2002)
Getzler, Ezra, Okounkov, Andrei, Pandharipande, Rahul
We give explicit formulas for the multipoint series of Gromov-Witten invariants of CP^1. We use the recursions of the Toda hierarchy to calculate these in degree 0, and the Toda equation and the...
Gromov-Witten theory, Hurwitz theory, and completed cycles (2002)
Okounkov, Andrei, Pandharipande, Rahul
We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The...
Generating functions for intersection numbers on moduli spaces of curves (2002)
Using the connection between intersection theory on the Deligne-Mumford spaces $${\overline{\mathcal{M}}}_{g,n}$$ and the edge scaling of the GUE matrix model, we express the n-point functions for...
A remark on Fourier pairing and binomial formula for Macdonald polynomials (2001)
We give a concise direct proof of the orthogonality of interpolation Macdonald polynomials with respect to the Fourier pairing and briefly discuss some immediate applications of this orthogonality,...
Okounkov, Andrei, Reshetikhin, Nikolai
Schur process is a time-dependent analog of the Schur measure on partitions studied in math.RT/9907127. Our first result is that the correlation functions of the Schur process are determinants with a...
Generating functions for intersection numbers on moduli spaces of curves (2001)
Using the connection between intersection theory on the Deligne-Mumford spaces and the edge scaling of the GUE matrix model (see math.CO/9903176, math.AG/0101147), we express the n-point functions...
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I (2001)
Okounkov, Andrei, Pandharipande, Rahul
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is...
We give a direct proof of the combinatorial formula for interpolation Macdonald polynomials by introducing certain polynomials, which we call generic Macdonald polynomials, which depend on $d$...
We compute the asymptotics of the number of connected branched coverings of a torus as their degree goes to infinity and the ramification type stays fixed. These numbers are equal to the volumes of...
Toda equations for Hurwitz numbers (2000)
We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings...
We give a representation-theoretic proof of the formula for correlation functions of z-measures obtained by Borodin and Olshanski in math.RT/9904010. This paper is historically preceding my paper...
Why would multiplicities be log-concave ? (2000)
It is a basic property of the entropy in statistical physics that is concave as a function of energy. The analog of this in representation theory would be the concavity of the logarithm of the...
Asymptotics of Plancherel measures for symmetric groups (2000)
Alexei Borodin, Andrei Okounkov, Grigori Olshanski
1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a...
Random matrices and random permutations (2000)
We prove the conjecture of Baik, Deift, and Johansson, which says that with respect to the Plancherel measure on the set of partitions λ of n, the rows$${\lambda }_{1},{\lambda }_{2},{\lambda...
A Fredholm determinant formula for Toeplitz determinants (1999)
Borodin, Alexei, Okounkov, Andrei
We prove a formula expressing a general n by n Toeplitz determinant as a Fredholm determinant of an operator 1-K acting on l_2({n,n+1,...}), where the kernel K admits an integral representation in...
Infinite wedge and random partitions (1999)
Using techniques from integrable systems, we obtain a number of exact results for random partitions. In particular, we prove a simple formula for correlation functions of what we call the Schur...
Asymptotics of Plancherel measures for symmetric groups (1999)
Borodin, Alexei, Okounkov, Andrei, Olshanski, Grigori
We consider the asymptotics of the Plancherel measures on partitions of $n$ as $n$ goes to infinity. We prove that the local structure of a Plancherel typical partition (which we identify with a...
Random Matrices and Random Permutations (1999)
We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions of $n$, the 1st, 2nd, and so on, rows behave, suitably scaled,...
On the representations of the infinite symmetric group (1998)
We classify all irreducible admissible representations of three Olshanski pairs connected to the infinite symmetric group. In particular, our methods yield two simple proofs of the classical Thoma's...
Asymptotics of Jack polynomials as the number of variables goes to infinity, q-alg/9709011 (1998)
Andrei Okounkov, Grigori Olshanski
Abstract. In this paper we study the asymptotic behavior of the Jack rational functions P (z 1; : : : ; zn; `) as the number of variables n and the signature grow to infinity. Our results generalize...
Shifted Schur functions (1998)
Andrei Okounkov, Grigori Olshanski
The classical algebra of symmetric functions has a remarkable deformation which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by...
A characterization of interpolation Macdonald polynomials (1997)
In this elementary paper we prove that the extra vanishing property characterizes the BC interpolation Macdonald polynomials inside a very general class of multivariate interpolation polynomials. It...
The Character of the Infinite Wedge Representation (1997)
Bloch, Spencer, Okounkov, Andrei
We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain...
Asymptotics of Jack polynomials as the number of variables goes to infinity (1997)
Okounkov, Andrei, Olshanski, Grigori
In this paper we study the asymptotic behavior of the Jack rational functions as the number of variables grows to infinity. Our results generalize the results of A. Vershik and S. Kerov obtained in...
The boundary of Young graph with Jack edge multiplicities (1997)
Kerov, Sergei, Okounkov, Andrei, Olshanski, Grigori
Consider the lattice of all Young diagrams ordered by inclusion, and denote by Y its Hasse graph. Using the Pieri formula for Jack symmetric polynomials, we endow the edges of the graph Y with formal...
On n-point correlations in the log-gas at rational temperature (1997)
We obtain a Ha type formula for n-point correlations in the log-gas at rational temperature (or, equivalently, n-point one-time ground state correlations in the quantum Calogero-Sutherland model for...
Okounkov, Andrei, Olshanski, Grigori
Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra...
BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials (1996)
We consider 3-parametric polynomials which replace the A-series interpolation Macdonald polynomials in the BC case. For these polynomials, we prove: an integral representation, a combinatorial...
Shifted Jack polynomials, binomial formula, and applications (1996)
Okounkov, Andrei, Olshanski, Grigori
In this note we prove an explicit binomial formula for Jack polynomials and discuss some applications of it.
Binomial formula for Macdonald polynomials (1996)
We prove a binomial formula for Macdonald polynomials and consider applications of it.
Shifted Schur Functions (1996)
Okounkov, Andrei, Olshanski, Grigori
The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a...
(Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula (1996)
We extend some results about shifted Schur functions to the general context of shifted Macdonald polynomials. We obtain two explicit formulas for these polynomials: a $q$-integral representation and...
Young Basis, Wick Formula, and Higher Capelli Identities (1996)
We prove Capelli type identities which involve the whole universal enveloping algebra $U(gl(n))$ and matrix elements of irreducible representations of the symmetric group. These identities generalize...
Quantum Immanants and Higher Capelli Identities (1996)
We consider remarkable central elements of the universal enveloping algebra of the general linear algebra which we call quantum immanants. We express them in terms of generators $E_{ij}$ and as...
THEORY OF SYMMETRIC GROUPS (1996)
The Erwin, Schrödinger International Boltzmanngasse, Andrei Okounkov, Anatoly Vershik, Andrei Okounkov, Anatoly Vershik
for Mathematical Physics,