## Fredholm determinants and the mKdV/sinh-Gordon hierarchies (1996)

Venue: | Comm. Math. Phys |

Citations: | 16 - 4 self |

### BibTeX

@ARTICLE{Tracy96fredholmdeterminants,

author = {Craig A. Tracy and Harold Widom},

title = {Fredholm determinants and the mKdV/sinh-Gordon hierarchies},

journal = {Comm. Math. Phys},

year = {1996},

pages = {9506006}

}

### OpenURL

### Abstract

For a particular class of integral operators K we show that the quantity φ: = log det(I + K) − log det(I − K) satisfies both the integrated mKdV hierarchy and the sinh-Gordon hierarchy. This proves a conjecture of Zamolodchikov. I.

### Citations

321 |
Quantum inverse scattering method and correlation functions
- Korepin, Bogoliubov, et al.
- 1993
(Show Context)
Citation Context ...um field-theoretic models. In the simplest of cases the differential equations are one of the Painlevé equations. Some, but by no means a complete set of, references to these further developments are =-=[2, 3, 4, 5, 9, 10, 12]-=-. The review paper [6] can be consulted for more examples of this connection. In recent work by the present authors on random matrices, techniques were developed that gave simple proofs of the connect... |

98 |
H.: Fredholm determinants, differential equations and matrix models
- Tracy, Widom
- 1994
(Show Context)
Citation Context ...um field-theoretic models. In the simplest of cases the differential equations are one of the Painlevé equations. Some, but by no means a complete set of, references to these further developments are =-=[2, 3, 4, 5, 9, 10, 12]-=-. The review paper [6] can be consulted for more examples of this connection. In recent work by the present authors on random matrices, techniques were developed that gave simple proofs of the connect... |

90 |
Density matrix of an impenetrable Bose gas and the fifth Painleve transcendent, Physica 1D
- Jimbo, Miwa, et al.
- 1980
(Show Context)
Citation Context ...um field-theoretic models. In the simplest of cases the differential equations are one of the Painlevé equations. Some, but by no means a complete set of, references to these further developments are =-=[2, 3, 4, 5, 9, 10, 12]-=-. The review paper [6] can be consulted for more examples of this connection. In recent work by the present authors on random matrices, techniques were developed that gave simple proofs of the connect... |

64 |
A.: Differential equations for quantum correlation functions
- Its, Izergin, et al.
- 1990
(Show Context)
Citation Context |

62 |
Spin-spin correlation functions for the two-dimensional Ising model: Exact theory in the scaling region
- Wu, McCoy, et al.
- 1976
(Show Context)
Citation Context ...etween certain Fredholm determinants and total systems of differential equations. This connection first appeared in work on the scaling limit of the 2-point correlation function in the 2D Ising model =-=[7, 11]-=- and the subsequent generalization to n-point correlations and holonomic quantum fields [8]. In applications the Fredholm determinants are either correlation functions or closely related to correlatio... |

46 |
Holonomic quantum fields
- Sato, Jimbo, et al.
- 1983
(Show Context)
Citation Context ...on first appeared in work on the scaling limit of the 2-point correlation function in the 2D Ising model [7, 11] and the subsequent generalization to n-point correlations and holonomic quantum fields =-=[8]-=-. In applications the Fredholm determinants are either correlation functions or closely related to correlations functions in various statistical mechanical or quantum field-theoretic models. In the si... |

20 |
Differential equations for sine-Gordon correlation functions at the free fermion point
- Bernard, Leclair
- 1994
(Show Context)
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15 |
Painlevé functions of the third
- McCoy, Tracy, et al.
- 1977
(Show Context)
Citation Context ...etween certain Fredholm determinants and total systems of differential equations. This connection first appeared in work on the scaling limit of the 2-point correlation function in the 2D Ising model =-=[7, 11]-=- and the subsequent generalization to n-point correlations and holonomic quantum fields [8]. In applications the Fredholm determinants are either correlation functions or closely related to correlatio... |

6 |
Systems of partial differential equations for a class of operator determinants
- Tracy, Widom
- 1995
(Show Context)
Citation Context |

6 |
Painleve III and 2D polymers
- Zamolodchikov
- 1994
(Show Context)
Citation Context |

5 |
Spin systems, statistical mechanics and Painlevé functions
- McCoy
- 1992
(Show Context)
Citation Context ... of cases the differential equations are one of the Painlevé equations. Some, but by no means a complete set of, references to these further developments are [2, 3, 4, 5, 9, 10, 12]. The review paper =-=[6]-=- can be consulted for more examples of this connection. In recent work by the present authors on random matrices, techniques were developed that gave simple proofs of the connection between a large cl... |

4 |
On a class of polynomials connected with the KdV equation
- Adler, Moser
- 1978
(Show Context)
Citation Context ...t2 2 ∂φ ) 1 ∂t1 − 10 ( ∂φ ) 2 ∂3 φ ∂t 3 1 + 6 ( ∂φ ) ∂t1 5 , ∂t5 ∂t1 etc. (In general there are constant factors on the left sides which can be removed by changes of scale in the time variables; e.g. =-=[1]-=-) To go in the other direction we introduce the inverse of the operator appearing in (2), which is given by (D 2 − 4 ∂φ ∂t1 −1 ∂φ D D) ∂t1 −1 = 1 2 (D−1 e 2φ D −1 e −2φ + D −1 e −2φ D −1 e 2φ ). (3) (... |