On the equilibrium measure for the Lukyanov integral

TL;DR

This paper rigorously constructs the equilibrium measure for a specific large-$N$ integral related to the Sinh-Gordon quantum field, supporting Lukyanov's conjecture on its asymptotic behavior.

Contribution

It provides a rigorous construction of the equilibrium measure and supports the predicted large-$N$ asymptotics of Lukyanov integrals, advancing understanding of quantum field expectations.

Findings

Construction of the equilibrium measure for the Lukyanov integral

Heuristic validation of the large-$N$ asymptotic exponent $\sigma$

Support for Lukyanov's conjectured power-law behavior

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 $N\rightarrow + \infty$.