Critical Gaussian multiplicative chaos for singular measures

Given $d\ge 1$, we provide a construction of the random measure – the critical Gaussian Multiplicative Chaos – formally defined as $e^{\sqrt{2d}X}d\mu$ where $X$ is a $log$-correlated Gaussian field and $\mu$ is a locally finite measure on $R^d$. Our construction generalizes the one performed in the case where $\mu$ is the Lebesgue measure. It requires that the measure $\mu$ is sufficiently spread out, namely that for $\mu$ almost every $x$ we have ${\int }_{B(x,1)}\frac{\mu \left(dy\right)}{{|x-y|}^d{e}^{\rho \left(log\frac{1}{|x-y|}\right)}}<\infty ,$ where $\rho :R_{+}\to R_{+}$ can be chosen to be any lower envelope function for the 3-Bessel process (this includes $\rho \left(x\right)=x^{\alpha }$ with $\alpha \in (0,1/2)$). We prove that three distinct random objects converge to a common limit which defines the critical GMC: the derivative martingale, the critical martingale, and the exponential of the mollified field. We also show that the above criterion for the measure $\mu$ is in a sense optimal.