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On the equilibrium measure for the Lukyanov integral

Charlie Dworaczek Guera, Karol K. Kozlowski

TL;DR

The paper rigorously constructs the $N$-dependent equilibrium measure governing the leading large-$N$ behavior of Lukyanov's integral ratio, providing a precise variational framework and an explicit Wiener-Hopf–Riemann–Hilbert representation for the equilibrium density. It proves the measure is Lebesgue continuous with a square-root vanishing density on a single interval, and derives explicit large-$N$ expansions for the endpoints. Using these results, it obtains the leading large-$N asymptotics of the interpolating integral, showing the Lukyanov ratio behaves as $(N/\mathfrak{r})^{\alpha^2/2}$ times a field-expectation factor, with a remainder that is presently bounded by $O(\sqrt{N}\,\tau_N^2)$ and conjectured to reduce to $O(1)$ via loop equations. Overall, the work provides a solid rigorous foundation for the conjectured large-$N structure and paves the way for a complete asymptotic expansion through analytic techniques.

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

On the equilibrium measure for the Lukyanov integral

TL;DR

The paper rigorously constructs the -dependent equilibrium measure governing the leading large- behavior of Lukyanov's integral ratio, providing a precise variational framework and an explicit Wiener-Hopf–Riemann–Hilbert representation for the equilibrium density. It proves the measure is Lebesgue continuous with a square-root vanishing density on a single interval, and derives explicit large- expansions for the endpoints. Using these results, it obtains the leading large-(N/\mathfrak{r})^{\alpha^2/2}O(\sqrt{N}\,\tau_N^2)O(1)N structure and paves the way for a complete asymptotic expansion through analytic techniques.

Abstract

In 2000, Lukyanov conjectured that a certain ratio of -fold integrals should provide access, in the large- regime, to the ground state expectation value of the exponential of the Sinh-Gordon quantum field in 1+1 dimensions and finite volume . This work aims at rigorously constructing the fundamental objects necessary to address the large- analysis of such integrals. More precisely, we construct and establish the main properties of the the equilibrium measure minimising a certain -dependent energy functional that naturally arises in the study of the leading large- behaviour of the Lukyanov integral. Our construction allows us to heuristically advocate the leading term in the large- asymptotic behaviour of the mentioned ratio of Lukyanov integrals, hence supporting Lukyanov's prediction -- obtained by other means -- on the exponent of the power-law term of its asymptotic expansion as .
Paper Structure (9 sections, 10 theorems, 188 equations, 1 figure)

This paper contains 9 sections, 10 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction and statement of results
  2. The separation of variable integral for the exponent of the field
  3. The main results
  4. The $N$-dependent equilibrium measure
  5. General properties of the equilibrium measure
  6. A truncated Wiener-Hopf based representation for the equilibrium measure
  7. A convenient representation for the inverse acting on $V^{\prime}_{N;\alpha}$
  8. Support of the equilibrium measure
  9. Large-$N$ behaviour of the interpolating integral

Key Result

Theorem 1.1

The equilibrium measure $\widehat{\mu}_{\mathrm{eq};\alpha}$. is Lebesgue continuous with a density $\widehat{\varrho}_{\mathrm{eq};\alpha}$ given by the square root of an analytic function. There exists $\mathfrak{r}_0$ such that, for any $N$ and $\mathfrak{r} \geq \mathfrak{r}_0$, it is supported where $\mathcal{W}_N$ is the integral transform given by ecriture transformation integrale WN in wh

Figures (1)

  • Figure 1: Contour $\Gamma_{\uparrow/\downarrow}$ delimiting regions of uniform asymptotic expansion of $\chi$.

Theorems & Definitions (10)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Lemma 2.8
  • Proposition 3.1