Critical threshold for weakly interacting log-correlated focusing Gibbs measures

2024/12/13 by Greco, Damiano, Oh, Tadahiro, Tao, Liying +1 · 2 citations
#35Q55 #60H30 #81T08 #82B26 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2412.09790

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 L²-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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• Probabilistic well-posedness of dispersive PDEs beyond variance blowup I: Benjamin-Bona-Mahony equation
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