13 problems

Take a polygon $\Omega$ whose boundary is composed of edges of a hexagonal lattice of mesh $h$, and let $A_1,A_2,A_3,A_4,A_5,A_6$ be boundary vertices. Assign each hexagon one of f…

Let $\kappa\in(0,8)$, let $\nu>(-4)\vee(\kappa/2-6)$, and let $(\Omega;x_1,x_2,x_3,x_4)$ be a quad. Let $\eta$ be an $\operatorname{hSLE}_{\kappa}(\nu)$ in $\Omega$ from $x_1$ to…

Let $\phi$ be the three-dimensional BMS field sampled according to the non-Gaussian measure $\nu_{\phi}$. A field is subordinated to a Gaussian field with finite Wiener chaos repre…

The pathwise square conjecture. For every test function $f$, the smeared fields converge in every $L^p(S'(\mathbb{R}^3),\nu_{\phi})$, $p\in1,\infty)$, and almost surely to…

Let $\phi$ be the BMS field and $N[\phi^2]$ its renormalized square, with scaling exponents $[\phi]$ and $[\phi^2]$. Let $c_{1,1;0}$ and $c_{1,1;2}$ be constants. The spin-spin OPE…

Let $\nu_{\phi\times\phi^2}$ be the joint law of the BMS field $\phi$ and its renormalized square $N[\phi^2]$, with the geometric symmetries acting diagonally on the two components…

The anomalous-square scaling conjecture. For small positive $\epsilon$, the pair $(\phi_{r,s},N_{r,s}[\phi^2])$ converges in distribution to a pair $(\phi,N[\phi^2])$ with joint la…

Let $\nu_{\phi}$ be the non-Gaussian BMS scaling-limit measure in $\mathbb{R}^3$, and consider critical lattice regularizations of the long-range model with the same scaling expone…

Let $\nu_{\phi}$ be the BMS fixed-point measure, and let $\mathcal{M}(\mathbb{R}^3)$ be the Möbius group of global conformal transformations of the compactified space…

Let $\nu_{\phi}$ be the non-Gaussian, translation-invariant, isotropic and scale-invariant BMS fixed-point measure on $S'(\mathbb{R}^3)$. The Osterwalder–Schrader positivity conjec…

Let $\nu_{\phi}$ be the non-Gaussian measure on $S'(\mathbb{R}^3)$ from the BMS fixed-point conjecture. A measure with moments of all orders is determined by pointwise correlations…

Let $L>1$, let $\epsilon>0$ be positive and small enough, and set … For the regularized measures $\nu_{r,s}$ defined by the ultraviolet and infrared cutoffs, use the bare parameter…

Generalized chordal SLE boundary duality conjecture. The right boundary $\partial_{\mathbb H}^+K(\infty)$ has the same distribution as the image of a chordal SLE…