论文标题
售价为$ 2D $库仑气和最大数量值高斯免费场的涡流波动的定量界限
Quantitative bounds on vortex fluctuations in $2d$ Coulomb gas and maximum of the integer-valued Gaussian free field
论文作者
论文摘要
在本文中，我们研究了涡旋对$ 2D $系统波动的影响，例如库仑气体，小人模型或整数值得高斯自由场。对于$ 2D $的小人模型，我们证明涡流引起的波动至少与自旋波产生的波动相同。我们在$ \ mathbb {z}^2 $中的两点相关性上获得以下定量上限，当$β> 1 $ \ [\langleσ_xσ_y\rangle_β^{villain} {villain} \ leq c \ leq c \ frac 1 {\ frac 1 { {2πβ}。此外，它提供了一种新的算法有效方法来取样$ 2D $库仑汽油。对于$ 2D $库仑气体，我们在高温下的波动上获得以下下限，\ [\ mathbb {e}_β^{coul} [\langleΔ^{ - 1} q，g \ rangle] \ rangle \]此估计与基于[JKKN77]的RG分析的预测相吻合，并表明在反温度$β$下的库仑电位$δ^{ - 1} q $应像阶数$ \ exp（π^2β）$的高斯自由温度的自由范围。最后，我们通过表明其最大偏差以定量的方式与通常的GFF的最大值来传递上述涡旋波动到整数值的GFF。更准确地说，我们表明，当$β> 1 $ \ [\ max_ {x \ in [-n，n，n]^2}ψ_n（x）\ leq \ sqrt {\ frac {2β}π\ big big big（1-βe^^^{ - \ frac { - \ frac） \，。 \]其中$ψ_n$是盒子$ [ - n，n]^2 $在反温度下$β^{ - 1} $的整数值GFF。还考虑了库仑气体，小人模型和整数值GFF的自由能量的应用。
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.
