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 $f,g\in C_c^{\mathrm{\infty }}(D)$ (smooth functions with compact support), we have that $$𝔼[⟨X,f⟩⟨X,g⟩]=\underset{D}{\int }\underset{D}{\int }f(x)g(y)C(x,y)dxdy$$where

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

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

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

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 $\varrho :D⟶ℝ$, define ${\varrho }_{\epsilon }(x)={\epsilon }^{-1}\varrho (x/\epsilon ).$ We define its regularisation as $X_{\epsilon }=X*{\varrho }_{\epsilon }$.

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

$\underset{\epsilon \to 0^{+}}{\lim }{e}^{\gamma X_{\epsilon }}{\epsilon }^{{\gamma }^2/2}$.

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