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arXiv · 2406.00145

On the equilibrium measure for the Lukyanov integral

Charlie Dworaczek Guera·Karol K. Kozlowski
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Abstract

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 $\sigma$ of the power-law $N^{\sigma}$ term of its asymptotic expansion as $N\rightarrow + \infty$.

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Charlie Dworaczek Guera, Karol K. Kozlowski. 2024-05-31. On the equilibrium measure for the Lukyanov integral. https://doi.org/10.1063/5.0252653

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