6 problems

Set $\alpha \defeq 1/2+4/\kappa$. Let $(\mathcal{D},\Phi,0,\infty)$ be a $\gamma$-Liouville quantum gravity disk, and let $\Gamma$ be an independent non-nested…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant probability measure…

Let $X$ be the carpet of a non-nested $\operatorname{CLE}_\kappa$. Given $X$, let $x$ and $y$ be independent samples from the canonical conformally covariant measure $\mu_X$, norma…

For $\kappa \in (8/3,4)$, let $\theta(\kappa)$ denote the distance exponent associated with the canonical geodesic metric on the $\operatorname{CLE}_\kappa$ carpet. Distance-expone…

Fix $s>1$, and let $(M_k',L_k')$ be sampled from the modified iteration procedure. Let $\mathbf c_{\mathrm L}\in(1,25)$ be the central charge parameter determined by $s$ through ……

For $k\in\mathbb N$, let $(M_k,L_k)$ be sampled from the probability measure on triangulations of the disk decorated by fully packed $O(2)$ loop configurations, at the critical par…