关于k-粗MBRW的Liouville热核和非普适性
On the Liouville heat kernel for k-coarse MBRW and nonuniversality
摘要：
本文研究了二维环面上的一类数值关联高斯场的Liouville热核（在$L^2$阶段）。我们发现，对于每个$\varepsilon>0$存在这样一个字段，其协方差是一个有界扰动的二维高斯自由场，并使相关的刘维热核满足短时间的估计，$$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 γ^2 } - \varepsilon } \right) \le p_t^γ(x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 γ^2 } + \varepsilon } \right) , $$为$γ<1/2$。特别是，这些是不同的预测，由于Watabiki，关于刘维热核的二维高斯自由场。
英文摘要：
We study the Liouville heat kernel (in the $L^2$ phase) associated with a class of logarithmically correlated Gaussian fields on the two dimensional torus. We show that for each $\varepsilon>0$ there exists such a field, whose covariance is a bounded perturbation of that of the two dimensional Gaussian free field, and such that the associated Liouville heat kernel satisfies the short time estimates, $$ \exp \left( - t^{ - \frac 1 { 1 + \frac 1 2 γ^2 } - \varepsilon } \right) \le p_t^γ(x, y) \le \exp \left( - t^{- \frac 1 { 1 + \frac 1 2 γ^2 } + \varepsilon } \right) , $$ for $γ<1/2$. In particular, these are different from predictions, due to Watabiki, concerning the Liouville heat kernel for the two dimensional Gaussian free field.
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