Critical threshold for weakly interacting log-correlated focusing Gibbs measures

Damiano Greco; Tadahiro Oh; Liying Tao; Leonardo Tolomeo

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arXiv · 2412.09790

Critical threshold for weakly interacting log-correlated focusing Gibbs measures

Damiano Greco·Tadahiro Oh·Liying Tao·Leonardo Tolomeo

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Abstract

We study log-correlated Gibbs measures on the $d$-dimensional torus with weakly interacting focusing quartic potentials whose coupling constants tend to $0$ as we remove regularization. In particular, we exhibit a phase transition for this model by identifying a critical threshold, separating the weakly and strongly coupling regimes; in the weakly coupling regime, we show that the frequency-truncated measures converge to the base Gaussian measure (possibly with a renormalized $L^2$-cutoff), whereas, in the strongly coupling regime, we prove non-convergence of the frequency-truncated measures, even up to a subsequence. Our result answers an open question posed by Brydges and Slade (1996).

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BibTeX RIS

Damiano Greco, Tadahiro Oh, Liying Tao, Leonardo Tolomeo. 2024-12-13. Critical threshold for weakly interacting log-correlated focusing Gibbs measures. https://arxiv.org/abs/2412.09790

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