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A remark on Gibbs measures with log-correlated Gaussian fields

Tadahiro Oh, Kihoon Seong, Leonardo Tolomeo

Abstract

We study Gibbs measures with log-correlated base Gaussian fields on the $d$-dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When $d = 2$, our argument provides an alternative proof of the non-normalizability result for the focusing $Φ^4_2$-measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein's inequality on $\mathbb{R}^d$. We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

A remark on Gibbs measures with log-correlated Gaussian fields

Abstract

We study Gibbs measures with log-correlated base Gaussian fields on the -dimensional torus. In the defocusing case, the construction of such Gibbs measures follows from Nelson's argument. In this paper, we consider the focusing case with a quartic interaction. Using the variational formulation, we prove non-normalizability of the Gibbs measure. When , our argument provides an alternative proof of the non-normalizability result for the focusing -measure by Brydges and Slade (1996). Furthermore, we provide a precise rate of divergence, where the constant is characterized by the optimal constant for a certain Bernstein's inequality on . We also go over the construction of the focusing Gibbs measure with a cubic interaction. In the appendices, we present (a) non-normalizability of the Gibbs measure for the two-dimensional Zakharov system and (b) the construction of focusing quartic Gibbs measures with smoother base Gaussian measures, showing a critical nature of the log-correlated Gibbs measure with a focusing quartic interaction.

Paper Structure

This paper contains 15 sections, 13 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Log-correlated Gibbs measures
  3. Dynamical problems associated with the log-correlated Gibbs measures
  4. Non-normalizability of the focusing Gibbs measure
  5. Gibbs measure with the cubic interaction
  6. Preliminary lemmas
  7. Deterministic estimates
  8. Tools from stochastic analysis
  9. Non-normalizability of the focusing Gibbs measure with the quartic interaction
  10. Variational formulation
  11. Proof of Theorem \ref{['THM:1']}
  12. Proofs of the auxiliary lemmas
  13. Construction of the Gibbs measure with the cubic interaction
  14. On the Gibbs measure for the two-dimensional Zakharov system
  15. Focusing quartic Gibbs measures with smoother Gaussian fields

Key Result

Theorem \oldthetheorem

Let $\lambda > 0$ and $k = 4$. Then, given any $K > 0$, we have where $R_N$ is the renormalized potential energy defined in Wick0 with $k = 4$. Moreover, the divergence rate of $Z_{K,N}$ is given by as $N\to \infty$. Here, $C_B$ is the optimal constant in Bernstein's inequality: where $P$ is the sharp Fourier projection onto the unit ball: and $\sigma_N$ is defined in sigma1. Moreover, we have

Theorems & Definitions (35)

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  • ...and 25 more