On the equilibrium measure for the Lukyanov integral
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 SinhGordon 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 powerlaw $N^{\sigma}$ term of its asymptotic expansion as $N\rightarrow + \infty$.
 Publication:

arXiv eprints
 Pub Date:
 May 2024
 DOI:
 10.48550/arXiv.2406.00145
 arXiv:
 arXiv:2406.00145
 Bibcode:
 2024arXiv240600145D
 Keywords:

 Mathematical Physics;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 82B23;
 81Q80;
 47B35;
 60F10
 EPrint:
 32 pages, 1 figures