**Integrable models with open boundaries**

I studied the correlation functions of the XXZ spin-1/2 chain subject to diagonal boundary conditions. This investigation was carried out in the framework of the algebraic Bethe ansatz (ABA) for open chains developped by E. Sklyanin in the late eighties. Toghther with my collaborators we have show that it is possible, just as in the case of the periodic boundary conditions, to represent the so-called elementary blocks at distance*m*from the boundary as*m*fold integrals. This work has been done in collaboration with*N. Kitanine, G. Niccoli, J.M. Maillet, N.A. Slavnov and V. Terras.*. We have described our approach to correlation functions of integrable models with open boundaries in the articles- N. Kitanine, G. Niccoli, J.M. Maillet, N.A. Slavnov and V. Terras,
*"Correlation functions of the open XXZ chain I"*, arxiv 07071995 - N. Kitanine, G. Niccoli, J.M. Maillet, N.A. Slavnov and V. Terras,
*"Correlation functions of the open XXZ chain II"*arxiv 0803.3305 , - K.K Kozlowski,
*"On the emptiness formation probability of the open XXZ spin-1/2 chain"*arxiv 0708.0433 , J. Stat. Mech. (2008) P02006 .

- N. Kitanine, G. Niccoli, J.M. Maillet, N.A. Slavnov and V. Terras,
**Asymptotics of correlators in 1D quantum integrable models**

Together with my collaborators we have shown that it is possible to represent the generating function of certain correlation functions of in all integrable models associated to the six-vertex R matrix as a contour integral around the soltuions of Bethe Ansatz equations. This so-called Master equation allowed to obtain a new type series representation for this generating function. We were able to extract the asymptotic behavior of this series by mapping each term of the series to a part of the asymptotic expansion of the Fredholm determinant of the generalized sine kernel. In particular, the asympotics of the series allowed to obtain the large*m*behavior of the ground state expectation value of the spin-spin correlation function &lang &sigma_{1}^{z}&sigma_{m}^{z}&rang in the massless regime of the XXZ chain Our computations confirmed the CFT predictions for this quantity. Moreover, we were able to shed a new light at the structure of the constants appearing in the leading asymptotics of &lang &sigma_{1}^{z}&sigma_{m}^{z}&rang . This work has been done in collaboration with*N. Kitanine, J.M. Maillet, N.A. Slavnov and V. Terras.*and is gathered in the articles- N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras,
*"On correlation functions of integrable models associated to the six-vertex R-matrix"*, arxiv 0611.1142 , J.Stat.Mech. 0701 (2007) P022. - N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras,
*"Asymptotic behavior of correlation functions of massless quantum integrable models"*,

### Asymptotics of integral operators

- N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras,
**The generalized sine kernel**

Together with*N. Kitanine, J.M. Maillet, N.A. Slavnov and V. Terras*have studied the*x*&rarr &infin behavior of the generalised sine kernel acting on the interval [-q;q]. This integral operator belongs to the class of competely integrable integable operators. The resolvent as well as the Fredholm determinant of such operators can be constructed by solving some matrix Riemann-Hilbert problems (RHP). It happens that these RHP's can be solved asymptotically by using the Deift-Zhou steepest descent method. It was possible to obtain the large*x*asymptotics of the Fredholm determinants of the generalized sine kernel. These asymptotics reporduced sevral known results, in particular those of Buslayev, McCoy and . In particular we were able to shed a new light on the oscillating term of the asymptotic expansion of the Fredholm determinant. The asymptotics of the Fredholm determinant allows us to derive the asymptotic behavior of some*m*fold integrals over [-q;q]^{m}involving the sine kernel versus any holomorphic function in*m*variables. Finally, we were also able to invert asymptotically truncated Wiener-Hopf symbols generated by holomorphic kernels. This work has been subject of the article- N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras,
*"Riemann-Hilbert approach to the study of the Generalized sine kernel"*,

- N. Kitanine, K. Kozlowski, J. M. Maillet, N. A. Slavnov, V. Terras,
**Determinants of truncated Wiener-Hopf**

There has been a long history of obtaining the leading asymptotic behavior of Fredholm determinants of truncated Wiener-hopf operators generated by symbols having Fischer-Hartwig type singularities. I have shown that it is possible to relate truncated Wiene-Hopf operators to some generalized sine kernels having singular kernels. These integral operatos are still analysable through the RHP formalism. The asymptotic solution of this RHP allowed to obtain the asymptotics of the Fredholm determinant. These asymptotics reproduced all the known results in the particular cases of parameters. Moreover, they filled in the missing parts between what was know for Toeplitz determinants and those of truncated Wiener-Hopf operators. Actually, the asymptotic behavior of Toeplitz matrices generated by singular symbols was obtained as a by product of my analysis. This result is gathered in- K.K Kozlowski,
*"Determinants of truncated Wiener-Hopf operators with Fischer-Hartwig singularities"*,

- K.K Kozlowski,