Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models
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2024
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| author | Biskup, Marek Huang, Haiyu |
| author_facet | Biskup, Marek Huang, Haiyu |
| contents | Given a square box $Λ_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{ϕ_x\colon x\inΛ_n\}$ with law proportional to ${\rm e}^{\frac12β(ϕ,Δ_nϕ)}\prod_{x\inΛ_n}ν({\rm d}ϕ_x)$, where $β>0$ is the inverse temperature, $Δ_n$ is a hierarchical Laplacian on $Λ_n$, and $ν$ is a non-degenerate $1$-periodic measure on $\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as $n\to\infty$, for the covariance $\langleϕ_xϕ_y\rangle$ and the fractional charge $\langle {\rm e}^{2π{\rm i}α(ϕ_x-ϕ_y)}\rangle$ in the subcritical $β<β_{\rm c}:=π^2/\log L$, critical $β=β_{\rm c}$ and slightly supercritical $β>β_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $β$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08964 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models Biskup, Marek Huang, Haiyu Probability Mathematical Physics 82B26, 82B27, 82B28 Given a square box $Λ_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{ϕ_x\colon x\inΛ_n\}$ with law proportional to ${\rm e}^{\frac12β(ϕ,Δ_nϕ)}\prod_{x\inΛ_n}ν({\rm d}ϕ_x)$, where $β>0$ is the inverse temperature, $Δ_n$ is a hierarchical Laplacian on $Λ_n$, and $ν$ is a non-degenerate $1$-periodic measure on $\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as $n\to\infty$, for the covariance $\langleϕ_xϕ_y\rangle$ and the fractional charge $\langle {\rm e}^{2π{\rm i}α(ϕ_x-ϕ_y)}\rangle$ in the subcritical $β<β_{\rm c}:=π^2/\log L$, critical $β=β_{\rm c}$ and slightly supercritical $β>β_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $β$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point. |
| title | Phase transition and critical behavior in hierarchical integer-valued Gaussian and Coulomb gas models |
| topic | Probability Mathematical Physics 82B26, 82B27, 82B28 |
| url | https://arxiv.org/abs/2412.08964 |