标题： 关于卢克亚诺夫积分的平衡测度 显示英文标题

标题： On the equilibrium measure for the Lukyanov integral

摘要： 2000年，Lukyanov猜想，某个$N$重积分的比值在大$N$范围内可以获取1+1维、有限体积$R$的Sinh-Gordon量子场的指数的基态期望值。 这项工作旨在严格构造解决此类积分的大$N$分析所需的基本对象。 更准确地说，我们构造并建立了最小化某种依赖于$N$的能量泛函的平衡测度的主要性质，该能量泛函自然出现在研究Lukyanov积分的大$N$行为的主导项时。 我们的构造使我们能够从启发式角度提出所述Lukyanov积分比值在大$N$渐近行为中的主要项，从而支持Lukyanov的预测——该预测是通过其他方法得到的——关于其渐近展开中幂律$N^{\sigma}$项的指数$\sigma$as$N\rightarrow + \infty$。 显示英文摘要

摘要： 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 $N\rightarrow + \infty$.