On the equilibrium measure for the Lukyanov integral
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909890785574912 |
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| author | Guera, Charlie Dworaczek Kozlowski, Karol K. |
| author_facet | Guera, Charlie Dworaczek Kozlowski, Karol K. |
| contents | In 2000, Lukyanov conjectured that a certain ratio of $N$-fold integrals should provide access, in the large-$N$ regime, to the ground state expectation value of the exponential of the Sinh-Gordon quantum field in 1+1 dimensions and finite volume $R$. This work aims at rigorously constructing the fundamental objects necessary to address the large-$N$ analysis of such integrals. More precisely, we construct and establish the main properties of the the equilibrium measure minimising a certain $N$-dependent energy functional that naturally arises in the study of the leading large-$N$ behaviour of the Lukyanov integral. Our construction allows us to heuristically advocate the leading term in the large-$N$ asymptotic behaviour of the mentioned ratio of Lukyanov integrals, hence supporting Lukyanov's prediction -- obtained by other means -- on the exponent $σ$ of the power-law $N^σ$ term of its asymptotic expansion as $N\rightarrow + \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the equilibrium measure for the Lukyanov integral Guera, Charlie Dworaczek Kozlowski, Karol K. Mathematical Physics Exactly Solvable and Integrable Systems 82B23, 81Q80, 47B35, 60F10 In 2000, Lukyanov conjectured that a certain ratio of $N$-fold integrals should provide access, in the large-$N$ regime, to the ground state expectation value of the exponential of the Sinh-Gordon quantum field in 1+1 dimensions and finite volume $R$. This work aims at rigorously constructing the fundamental objects necessary to address the large-$N$ analysis of such integrals. More precisely, we construct and establish the main properties of the the equilibrium measure minimising a certain $N$-dependent energy functional that naturally arises in the study of the leading large-$N$ behaviour of the Lukyanov integral. Our construction allows us to heuristically advocate the leading term in the large-$N$ asymptotic behaviour of the mentioned ratio of Lukyanov integrals, hence supporting Lukyanov's prediction -- obtained by other means -- on the exponent $σ$ of the power-law $N^σ$ term of its asymptotic expansion as $N\rightarrow + \infty$. |
| title | On the equilibrium measure for the Lukyanov integral |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems 82B23, 81Q80, 47B35, 60F10 |
| url | https://arxiv.org/abs/2406.00145 |