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| Natura: | Preprint |
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2023
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| Accesso online: | https://arxiv.org/abs/2307.14904 |
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| _version_ | 1866914979280584704 |
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| author | Eynard, Bertrand Mitsios, Dimitrios Oukassi, Soufiane |
| author_facet | Eynard, Bertrand Mitsios, Dimitrios Oukassi, Soufiane |
| contents | An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $Ψ$ living in a Lie group $G$, which satisfies some differential equations with rational coefficients. From this wave function, it is usual to define a sequence of ``correlators" $W_n$, that play an important role in many applications in mathematical physics. Here, we show how to systematically obtain ordinary differential equations (ODE) and recursion relations with polynomial coefficients for the correlators.
An application is random matrix theory, where the wave functions are the expectation value of the characteristic polynomial, they form a family of orthogonal polynomials, and are known to satisfy an integrable system. The correlators are then the correlation functions of resolvents or of eigenvalue densities. We give the ODE and recursion on the matrix size that they satisfy. In addition, we discuss generic Fuchsian systems, namely, Schlesinger systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_14904 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Recursions and ODEs for the correlators in integrable systems and random matrices Eynard, Bertrand Mitsios, Dimitrios Oukassi, Soufiane Mathematical Physics High Energy Physics - Theory 17B80, 15B52, 53C05 An integrable system is often formulated as a flat connection, satisfying a Lax equation. It is given in terms of compatible systems having a common solution called the ``wave function" $Ψ$ living in a Lie group $G$, which satisfies some differential equations with rational coefficients. From this wave function, it is usual to define a sequence of ``correlators" $W_n$, that play an important role in many applications in mathematical physics. Here, we show how to systematically obtain ordinary differential equations (ODE) and recursion relations with polynomial coefficients for the correlators. An application is random matrix theory, where the wave functions are the expectation value of the characteristic polynomial, they form a family of orthogonal polynomials, and are known to satisfy an integrable system. The correlators are then the correlation functions of resolvents or of eigenvalue densities. We give the ODE and recursion on the matrix size that they satisfy. In addition, we discuss generic Fuchsian systems, namely, Schlesinger systems. |
| title | Recursions and ODEs for the correlators in integrable systems and random matrices |
| topic | Mathematical Physics High Energy Physics - Theory 17B80, 15B52, 53C05 |
| url | https://arxiv.org/abs/2307.14904 |