Draft:Gaussian Multiplicative Chaos

In Mathematics and Physics, Gaussian Multiplicative Chaos refers to a random measure obtained by the exponentiation of a log-correlated Gaussian field. Gaussian multiplicative chaos can be seen as a generalisation of Multiplicative cascade.

The most famous example is the so-called Liouville Quantum Gravity which can be understood by the limit of the exponential of a $2$-dimensional Gaussian free field in a bounded domain $D$.

Assume that $X$ is a random variable taking values within distributions on $D$. We say that such field is log-correlated if for any functions $f,g\in C_{c}^{\infty }(D)$ (smooth functions with compact support), we have that $$\mathbb {E} [\langle X,f\rangle \langle X,g\rangle ]=\int _{D}\int _{D}f(x)g(y)C(x,y)dxdy$$where

$C(x,y):=\gamma \max\{\log 1/|x-y|,0\}+e(x,y),$

for some positive constant and $e:D\times D\longrightarrow \mathbb {R}$ is a bounded function. Due to the fact that

$\lim _{x\to y}C(x,y)=\infty$,

we have that $X$ cannot be considered a function. Therefore, it is useful to define a regularisation of $X$, say, via mollification. That is, let $\varrho :D\longrightarrow \mathbb {R}$, define $\varrho _{\varepsilon }(x)=\varepsilon ^{-1}\varrho (x/\varepsilon ).$ We define its regularisation as $X_{\varepsilon }=X*\varrho _{\varepsilon }$.

We then define the $\gamma$-Gaussian Multiplicative Chaos of as the limit (as a measure) of the approximation

$\lim _{\varepsilon \to 0^{+}}e^{\gamma X_{\varepsilon }}\varepsilon ^{\gamma ^{2}/2}$.

The necessity of the term $\varepsilon ^{\gamma ^{2}/2}$is to