Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916250251165696 |
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| author | Garban, Christophe |
| author_facet | Garban, Christophe |
| contents | The Discrete Gaussian Chain is a model of interfaces $Ψ: \mathbf{Z} \to \mathbf{Z}$ governed by the Hamiltonian $$ H(Ψ)= \sum_{i\neq j} J_α(|i-j|) |Ψ_i -Ψ_j|^2 $$ with long-range coupling constants $J_α(k)\asymp k^{-α}$. For any $α\in [2,3)$ and at high enough temperature, we prove an invariance principle for such an $α$-Discrete Gaussian Chain towards a $H(α)$-fractional Gaussian process where the Hurst index $H$ satisfies $H=H(α)=\frac {α-2} 2$. This result goes beyond a conjecture by Fröhlich and Zegarlinski [FZ91] which conjectured fluctuations of order $n^{\tfrac 1 2 (α-2) \wedge 1}$ for the Discrete Gaussian Chain.
More surprisingly, as opposed to the case of the $2D$ Discrete Gaussian $Ψ: \mathbf{Z}^2 \to \mathbf{Z}$, we prove that the integers do not affect the {\em effective temperature} of the discrete Gaussian Chain at large scales. Such an {\em invisibility of the integers} had been predicted by Slurink and Hilhorst in the special case $α_c=2$ in [SH83]. We also identify a similar invisibility of integers when a $2D$ Gaussian Free Field at high temperature is conditioned to take integer values on a dilute enough "fractal subset" of $\mathbf{Z}^2$.
Our proof relies on four main ingredients:
(1) A Caffareli-Silvestre extension for the discrete fractional Laplacian (which may be of independent interest)
(2) A localisation of the chain in a smoother sub-domain
(3) A Coulomb gas-type expansion in the spirit of Fröhlich-Spencer [FS82]
(4) Controlling the amount of Dirichlet Energy supported by a $1D$ band for the Green functions of $\mathbf{Z}^2$ Bessel-type random walks
Finally, we also analyse the (easier) regime $α\in(1,2) \cup (3,\infty)$ as well as the $2D$ Discrete Gaussian with long-range coupling constants (for any $α>α_c=4$). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04536 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian Garban, Christophe Probability Mathematical Physics The Discrete Gaussian Chain is a model of interfaces $Ψ: \mathbf{Z} \to \mathbf{Z}$ governed by the Hamiltonian $$ H(Ψ)= \sum_{i\neq j} J_α(|i-j|) |Ψ_i -Ψ_j|^2 $$ with long-range coupling constants $J_α(k)\asymp k^{-α}$. For any $α\in [2,3)$ and at high enough temperature, we prove an invariance principle for such an $α$-Discrete Gaussian Chain towards a $H(α)$-fractional Gaussian process where the Hurst index $H$ satisfies $H=H(α)=\frac {α-2} 2$. This result goes beyond a conjecture by Fröhlich and Zegarlinski [FZ91] which conjectured fluctuations of order $n^{\tfrac 1 2 (α-2) \wedge 1}$ for the Discrete Gaussian Chain. More surprisingly, as opposed to the case of the $2D$ Discrete Gaussian $Ψ: \mathbf{Z}^2 \to \mathbf{Z}$, we prove that the integers do not affect the {\em effective temperature} of the discrete Gaussian Chain at large scales. Such an {\em invisibility of the integers} had been predicted by Slurink and Hilhorst in the special case $α_c=2$ in [SH83]. We also identify a similar invisibility of integers when a $2D$ Gaussian Free Field at high temperature is conditioned to take integer values on a dilute enough "fractal subset" of $\mathbf{Z}^2$. Our proof relies on four main ingredients: (1) A Caffareli-Silvestre extension for the discrete fractional Laplacian (which may be of independent interest) (2) A localisation of the chain in a smoother sub-domain (3) A Coulomb gas-type expansion in the spirit of Fröhlich-Spencer [FS82] (4) Controlling the amount of Dirichlet Energy supported by a $1D$ band for the Green functions of $\mathbf{Z}^2$ Bessel-type random walks Finally, we also analyse the (easier) regime $α\in(1,2) \cup (3,\infty)$ as well as the $2D$ Discrete Gaussian with long-range coupling constants (for any $α>α_c=4$). |
| title | Invisibility of the integers for the discrete Gaussian chain via a Caffarelli-Silvestre extension of the discrete fractional Laplacian |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2312.04536 |