## Infinite wedge and random partitions

Venue: | Selecta Mathematica (new series |

Citations: | 56 - 6 self |

### BibTeX

@ARTICLE{Okounkov_infinitewedge,

author = {Andrei Okounkov},

title = {Infinite wedge and random partitions},

journal = {Selecta Mathematica (new series},

year = {},

pages = {9907127}

}

### OpenURL

### Abstract

The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,

### Citations

840 |
Symmetric functions and Hall polynomials
- Macdonald
- 1979
(Show Context)
Citation Context ...unctions of Schur measure 2.1 The Schur measure 2.1.1 Definition Consider the set of all partitions λ and introduce the following function of λ M(λ) = 1 Z sλ(x) sλ(y) where sλ are the Schur functions =-=[24]-=- in auxiliary variables x1, x2, . . . and y1, y2, . . ., and Z is the sum in the Cauchy identity for the Schur functions Z = ∑ sλ(x) sλ(y) = ∏ (1 − xiyj) −1 . λ It is clear that if, for exmaple, {yi} ... |

726 |
Infinite Dimensional Lie Algebras
- Kac
- 1990
(Show Context)
Citation Context ...te wedge: summary of formulas In this section we collected, for the reader’s convenience, some basic formulas related to the infinite wedge space. This material is standard and Chapter 14 of the book =-=[20]-=- can be recommended as a reference. With a few exceptions, we are closely following [20]. Definition Let V be a linear space with basis {k}, k ∈ Z + 1 2 by definition, spanned by vectors vS = s1 ∧ s2 ... |

637 |
Random Matrices
- Mehta
- 1991
(Show Context)
Citation Context ...nel K has a nice generating function, and thus a contour integral representation, in terms of the parameters of the Schur measure M, see Theorem 2. Note the similarity to situation in random matrices =-=[25]-=-, but also observe that K does not involve any objects even remotely as complicated as polynomials orthogonal with respect to an arbitrary measure. We also prove that, as functions of the parameters o... |

345 | On the distribution of the length of the longest increasing subsequence in a random permutation
- Baik, Deift, et al.
- 1999
(Show Context)
Citation Context ...n, for example, in the case of random matrices. In a sense, we show that solitaire (which is related to increasing subsequences in random permutations and thus to the Plancherel measure on partitions =-=[2, 3]-=-) and soliton have much more in common than the general notion of solitude. The other character in the title, the infinite wedge space, is our main technical tool. Starting from the fundamental work o... |

345 | Combinatorial Enumeration - Goulden, Jackson - 1983 |

194 |
Theory of nonlinear lattices
- Toda
- 1988
(Show Context)
Citation Context ... α with a ± sign according to its height. Also remark that in the Plancherel case the correlations depend only on the product t1t ′ 1 and so (A.21) becomes an ODE equivalent to the usual Toda lattice =-=[30]-=-. 2.3.3 Other algebras and Kerov’s construction Instead of the operators αn, which generate an action of the Heisenberg algebra, one can consider representation of other algebras on Λ ∞ 2 V . Sandwich... |

136 | Longest increasing subsequences: from patience sorting to the Baik-DeiftJohansson theorem
- ALDOUS, P
- 1999
(Show Context)
Citation Context ...n, for example, in the case of random matrices. In a sense, we show that solitaire (which is related to increasing subsequences in random permutations and thus to the Plancherel measure on partitions =-=[2, 3]-=-) and soliton have much more in common than the general notion of solitude. The other character in the title, the infinite wedge space, is our main technical tool. Starting from the fundamental work o... |

64 |
A.: Differential equations for quantum correlation functions
- Its, Izergin, et al.
- 1990
(Show Context)
Citation Context ...factors after taking the partial with respect to t1. Similar formulas hold for partials with respect to other parameters. 92.2.4 K as an integrable kernel Recall that a kernel K is called integrable =-=[18, 11]-=- if the kernel (x−y) K(x, y) has finite rank. In more invariant terms, K is integrable, if its commutator with the operator of multiplication by the independent coordinate has finite rank. In our case... |

63 | Harmonic analysis on the infinite symmetric group
- Kerov, Olshanski, et al.
- 2004
(Show Context)
Citation Context ...hanski [10] for the correlation functions of the so called zmeasures, see Section 2.1.4. The asymptotics of z-measures is important for the harmonic analysis on the infinite symmetric group S(∞), see =-=[9, 23]-=-. Since our kernel K has a simple integral representation, the formula (1.1) is particularly suitable for asymptotic investigations. We also point out that our proofs are considerably simpler and argu... |

62 | Random matrices and random permutations - Okounkov - 1999 |

60 |
Statistical mechanics of combinatorial partitions and their limit shapes, Funct Anal Appl 30
- Vershik
- 1996
(Show Context)
Citation Context ...this formula can be also found by the Laplace method. This theorem says that locally a typical random partition is just a trajectory of a random walk. Theorem 7 is in agreement with Vershik’s theorem =-=[34]-=- about the limit shape of a typical partition with respect to the uniform measure. Namely, Vershik’s theorem asserts that after the scaling by the square root of the area in both direction, a typical ... |

53 |
Mirror symmetry and elliptic curves, The moduli space of curves (Texel Island
- Dijkgraaf
- 1994
(Show Context)
Citation Context ... for alternative proofs of the result of [7]. Again, (1.2) seems to be particularly suitable for asymptotic analysis. 1.4.3 Ramified coverings, moduli spaces, and ergodic theory It is known, see e.g. =-=[12]-=-, that the Schur and uniform measures on partitions are related to the enumeration of ramified coverings of the sphere and the torus, respectively. In the case of the torus, the exact formula (3.2) wa... |

51 | A Fredholm determinant formula for Toeplitz determinants
- Borodin, Okounkov
(Show Context)
Citation Context ...r proofs are considerably simpler and arguably much more conceptual than the ones given in [8, 10]. 1.4.2 Toeplitz determinants and Fredholm determinants The results of the present paper were used in =-=[7]-=- to solve a problem proposed, independently, by A. Its and P. Deift. The problem was to find a general identity of the form det(φi−j)1≤i,j≤n = det(1 − Kφ)ℓ2({n,n+1,... }) , (1.2) where the kernel Kφ i... |

48 | Distributions on partitions, point processes, and the hypergeometric kernel
- Borodin, Olshanski
- 2000
(Show Context)
Citation Context ...rmutation and also used in [8] to analyze the local structure of a Plancherel typical partition in the “bulk” of the limit shape. It also generalizes the more general formula of Borodin and Olshanski =-=[10]-=- for the correlation functions of the so called zmeasures, see Section 2.1.4. The asymptotics of z-measures is important for the harmonic analysis on the infinite symmetric group S(∞), see [9, 23]. Si... |

40 | Point processes and the infinite symmetric group
- BORODIN, OLSHANSKI
- 1998
(Show Context)
Citation Context ...hanski [10] for the correlation functions of the so called zmeasures, see Section 2.1.4. The asymptotics of z-measures is important for the harmonic analysis on the infinite symmetric group S(∞), see =-=[9, 23]-=-. Since our kernel K has a simple integral representation, the formula (1.1) is particularly suitable for asymptotic investigations. We also point out that our proofs are considerably simpler and argu... |

33 |
Polynomial functions on the set of Young diagrams
- Kerov, Olshanski
- 1994
(Show Context)
Citation Context ...nd provides another proof of the Borodin-Olshanski formula, see [29]. The action of sl2 in the basis {vλ} gives certain operators on partitions considered by S. Kerov in his analysis of the z-measure =-=[21]-=-. 3 The uniform measure In this section we shall consider the uniform measure on partitions on n and the related measure Mq(λ) = (q; q)∞ q |λ| , q ∈ [0, 1) , on the set of all partitions. The normaliz... |

27 | Integrals over Classical Groups, Random Permutations, Toda and Toeplitz Lattices
- Adler, Moerbeke
(Show Context)
Citation Context .... . . } , 2 these τ-functions are Fredholm determinants with the kernel K. The Toda lattice equations for these Fredholm determinants are related to the results of 11[31, 32] and of the recent paper =-=[1]-=-. See [7] for a discussion of the relationship between these Fredholm determinants and Toeplitz determinants. The times t and t ′ in the Toda lattice hierarchy admit the following combinatorial interp... |

21 | The character of the infinite Wedge representation
- Bloch, Okounkov
(Show Context)
Citation Context ...tforward once one interprets the correlation functions as certain matrix elements in the infinite wedge space. 1.3 Uniform measure We give a new, more simple, and conceptual proof of the formula from =-=[5]-=-, which is reproduced in Theorem 4 below, for the following averages, called the n-point functions, F(t1, . . ., tn) = ∑ q |λ| λ n∏ ∞∑ 1 λi−i+ 2 tk . k=1 i=1 These n-point functions are sums of determ... |

19 | On a Toeplitz determinant identity of Borodin and Okounkov
- BASOR, WIDOM
(Show Context)
Citation Context .... }) , (1.2) where the kernel Kφ in the Fredholm determinant admits an integral representation in terms of the generating function φ(z) = ∑ φnz n for the entries of the Toeplitz determinant. See also =-=[4]-=- for alternative proofs of the result of [7]. Again, (1.2) seems to be particularly suitable for asymptotic analysis. 1.4.3 Ramified coverings, moduli spaces, and ergodic theory It is known, see e.g. ... |

18 |
lattice hierarchy, Adv
- Ueno, Takasaki
- 1984
(Show Context)
Citation Context ... measure. We also prove that, as functions of the parameters of the measure M, the correlation functions satisfy an infinite hierarchy of PDE’s, namely the Toda lattice hierarchy of Ueno and Takasaki =-=[33]-=-, see Theorem 3. Again, this is a well known phenomenon in mathematical physics that various correlation functions tend to be τ-functions of integrable hierarchies. Both of these results are quite str... |

13 |
Asymptotic formulas on flat surfaces
- Eskin, Masur
(Show Context)
Citation Context ...ree goes to infinity. These numbers, which are certain rather complicated polynomials in Bernoulli numbers, can be identified with volumes of certain moduli spaces and are important in ergodic theory =-=[14]-=-. For the coverings of the sphere, see [28]. 31.5 Acknowledgments I very much benefited from the discussions with S. Bloch, A. Borodin, P. Deift, A. Eskin, M. Kashiwara, S. Kerov, G. Olshanski, Ya. P... |

10 |
Discrete orthogonal polynomial and the Plancherel measure
- Johansson
- 2001
(Show Context)
Citation Context ...pplications 21.4.1 Asymptotic problems Our formula for the correlation functions of the Schur measure generalizes the exact formulas for the correlations functions for the Plancherel measure used in =-=[8, 19]-=- to prove the conjecture of Baik, Deift, and Johansson [3] about increasing subsequences in a random permutation and also used in [8] to analyze the local structure of a Plancherel typical partition i... |

7 |
Integrable operators
- Deift
- 1999
(Show Context)
Citation Context ...factors after taking the partial with respect to t1. Similar formulas hold for partials with respect to other parameters. 92.2.4 K as an integrable kernel Recall that a kernel K is called integrable =-=[18, 11]-=- if the kernel (x−y) K(x, y) has finite rank. In more invariant terms, K is integrable, if its commutator with the operator of multiplication by the independent coordinate has finite rank. In our case... |

6 | The theory of determinants, 2nd edition - Muir - 1906 |

3 | The mathematical pamphlets of Charles Lutwidge Dodgson and related pieces - Dodgson - 1994 |

2 |
Branched coverings of the torus and volumes of spaces of Abelian differentials
- Eskin, Okounkov
(Show Context)
Citation Context ... Schur and uniform measures on partitions are related to the enumeration of ramified coverings of the sphere and the torus, respectively. In the case of the torus, the exact formula (3.2) was used in =-=[15]-=- to compute the asymptotics of the number of ramified coverings with given ramification type as the degree goes to infinity. These numbers, which are certain rather complicated polynomials in Bernoull... |

2 |
Toda equations for Hurwitz numbers, preprint (math.AG/0004128
- Okounkov
(Show Context)
Citation Context ...are certain rather complicated polynomials in Bernoulli numbers, can be identified with volumes of certain moduli spaces and are important in ergodic theory [14]. For the coverings of the sphere, see =-=[28]-=-. 31.5 Acknowledgments I very much benefited from the discussions with S. Bloch, A. Borodin, P. Deift, A. Eskin, M. Kashiwara, S. Kerov, G. Olshanski, Ya. Pugaj, T. Spencer, A. Vershik and others. Mu... |