二维库仑气体中涡旋涨落的定量界与整值高斯自由场的最大值
Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field
AI总结：
本文研究二维库仑气体、Villain模型及整值高斯自由场中涡旋对涨落的影响，给出定量上下界，证明涡旋效应与自旋波同阶，并提出高效采样方法及对偶转移结果。
摘要：
本文研究了涡旋对二维系统（如库仑气体、Villain模型或整值高斯自由场）涨落的影响。对于二维Villain模型，我们证明了由涡旋引起的涨落至少与自旋波产生的涨落具有相同的数量级。当β>1时，我们得到了在Z²中两点相关函数的如下定量上界：\\[ \langleσ_x σ_y\rangle_β^{Villain} \leq C \\, \left( \frac 1 {\\|x-y\\|_2}\right)^{\frac 1 {2πβ}\left ( 1+βe^{-\frac{(2π)^2}{2} β}\right )} \\] 证明完全是非微扰的。此外，它提供了一种新的且算法上高效的采样二维库仑气体的方法。对于二维库仑气体，我们在高逆温度下得到其涨落的如下下界：\\[ \mathbb{E}_β^{Coul}[\langle Δ^{-1}q, g\rangle] \geq \exp(-π^2 β+ o(β)) \langle g,(-Δ)^{-1}g \rangle \\] 该估计与基于[JKKN77]中RG分析得到的预测一致，并表明在逆温度β下，库仑势Δ^{-1}q应像逆温度数量级为exp(π^2 β)的高斯自由场那样缩放。最后，我们通过对偶恒等式将上述涡旋涨落转移到整值GFF，表明其最大值在定量上偏离通常GFF的最大值。更精确地，我们证明当β>1时，以高概率有：\\[ \max_{x\in [-n,n]^2} Ψ_n(x) \leq \sqrt{\frac{2β}π \big(1 - βe^{- \frac{(2π)^2β} {2} } \big)} \log n \\,. \\] 其中Ψ_n是盒子[-n,n]²中在逆温度β^{-1}下的整值GFF。还考虑了这些结果对库仑气体、Villain模型和整值GFF自由能的应用。
英文摘要：
In this paper, we study the influence of the vortices on the fluctuations of $2d$ systems such as the Coulomb gas, the Villain model or the integer-valued Gaussian free field. In the case of the $2d$ Villain model, we prove that the fluctuations induced by the vortices are at least of the same order of magnitude as the ones produced by the spin-wave. We obtain the following quantitative upper-bound on the two-point correlation in $\mathbb{Z}^2$ when $β>1$ \[ \langleσ_x σ_y\rangle_β^{Villain} \leq C \, \left( \frac 1 {\|x-y\|_2}\right)^{\frac 1 {2πβ}\left ( 1+βe^{-\frac{(2π)^2}{2} β}\right )} \] The proof is entirely non-perturbative. Furthermore it provides a new and algorithmically efficient way of sampling the $2d$ Coulomb gas. For the $2d$ Coulomb gas, we obtain the following lower bound on its fluctuations at high inverse temperature \[ \mathbb{E}_β^{Coul}[\langle Δ^{-1}q, g\rangle] \geq \exp(-π^2 β+ o(β)) \langle g,(-Δ)^{-1}g \rangle \] This estimate coincides with the predictions based on a RG analysis from [JKKN77] and suggests that the Coulomb potential $Δ^{-1}q$ at inverse temperature $β$ should scale like a Gaussian free field of inverse temperature of order $\exp(π^2 β)$. Finally, we transfer the above vortex fluctuations via a duality identity to the integer-valued GFF by showing that its maximum deviates in a quantitative way from the maximum of a usual GFF. More precisely, we show that with high probability when $β>1$ \[ \max_{x\in [-n,n]^2} Ψ_n(x) \leq \sqrt{\frac{2β}π \big(1 - βe^{- \frac{(2π)^2β} {2} } \big)} \log n \,. \] where $Ψ_n$ is an integer-valued GFF in the box $[-n,n]^2$ at inverse temperature $β^{-1}$. Applications to the free-energies of the Coulomb gas, the Villain model and the integer-valued GFF are also considered.
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