Equivalence of Liouville measure and Gaussian free field
ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES(2014)
摘要
Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity measure $\mu := e^{\gamma h(z)} dz.$ It is known that the field $h$ a.s. determines the measure $\mu_h$. We show that the converse is true: namely, $h$ is measurably determined by $\mu_h$. More generally, given a random closed fractal subset $X$ endowed with a Frostman measure $\sigma_X$ whose support is $X$ (independent of $h$), we construct a quantum measure $\mu_X$ and ask the following: how much information does $\mu_X$ contain about the free field? We conjecture that $\mu_X$ always determines $h$ restricted to $X$, in the sense that it determines its harmonic extension off $X$. We prove the conjecture in the case where $X$ is an independent SLE$_\kappa$ curve equipped with its quantum natural time, and in the case where $X$ is Liouville Brownian motion (that is, standard Brownian motion equipped with its quantum clock). The proof in the latter case relies on properties of nonintersecting planar Brownian motion, including the value of some nonintersection exponents.
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关键词
Gaussian free field, Gaussian multiplicative chaos, Liouville measure
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