We study the extremal process associated with the Discrete Gaussian Free Field on the square lattice and elucidate how the conformal symmetries manifest themselves in the scaling limit. Specifically, we prove that the joint process of spatial positions ( x ) and centered values ( h ) of the extreme local maxima in lattice versions of a bounded domain $$D\subset {\mathbb {C}}$$ D ⊂ C converges, as the lattice spacing tends to zero, to a Poisson point process with intensity measure $$Z^D(\mathrm{d}x)\otimes \mathrm{e}^{-\alpha h}\mathrm{d}h$$ Z D ( d x ) ⊗ e - α h d h , where $$\alpha $$ α is a constant and $$Z^D$$ Z D is a random a.s.-finite measure on D . The random measures $$\{Z^D\}$$ { Z D } are naturally interrelated; restrictions to subdomains are governed by a Gibbs–Markov property and images under analytic bijections f by the transformation rule $$(Z^{f(D)}\circ f)(\mathrm{d}x)\,\overset{\mathrm{law}}{=}\,|f'(x)|^4\, Z^D(\mathrm{d}x)$$ ( Z f ( D ) ∘ f ) ( d x ) = law | f ′ ( x ) | 4 Z D ( d x ) . Conditions are given that determine the laws of these measures uniquely. These identify $$Z^D$$ Z D with the critical Liouville Quantum Gravity associated with the Continuum Gaussian Free Field.

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二维离散高斯自由场极值过程中的共形对称性

我们研究了与方形晶格上的离散高斯自由场相关的极值过程，并阐明了共形对称性如何在标度限制中表现出来。具体来说，我们证明了在有界域 $$D\subset {\mathbb {C}}$$ D ⊂ C 的格子版本中，空间位置 ( x ) 和极局部最大值的中心值 ( h ) 的联合过程收敛，随着晶格间距趋于零，到具有强度度量的泊松点过程 $$Z^D(\mathrm{d}x)\otimes \mathrm{e}^{-\alpha h}\mathrm{d}h $$ ZD ( dx ) ⊗ e - α hdh ，其中 $$\alpha $$ α 是常数，$$Z^D$$ ZD 是 D 上的随机有限测度。随机度量 $$\{Z^D\}$$ { ZD } 自然是相互关联的；对子域的限制由 Gibbs–Markov 属性控制，解析双射 f 下的图像由转换规则 $$(Z^{f(D)}\circ f)(\mathrm{d}x)\,\overset{\ mathrm{law}}{=}\,|f'(x)|^4\, Z^D(\mathrm{d}x)$$ ( Z f ( D ) ∘ f ) ( dx ) = law | f ′ ( x ) | 4 ZD (dx)。给出了唯一确定这些措施的法律的条件。这些将 $$Z^D$$ZD 与与连续高斯自由场相关的临界刘维尔量子引力识别。