16 problems

Let $\kappa>4$ and set $\kappa'=16/\kappa$. In the configuration $(\mathbb{H},x,y,z,\infty)$ with $x<y<z$, let $(K_t)$ be the Loewner chain obtained by running an…

Let $D$ be a domain with boundary points $A$ and $B$, and let $P_{D,A,B}^{SAW}$ denote a scaling limit of measures on self-avoiding curves from $A$ to $B$ in $D$, assigning weight…

Let the c3-finite bead process be conditioned to survive up to distance $r$, equivalently up to time $r^2$ or imaginary part $r$, and let $r\rightarrow\infty$. Scaling limit conjec…

Consider the loop $O(n)$ model with $n\in (0,2]$ and boundary condition $\beta\in\mathrm{LP}_N$. Let $\varsigma^{\delta}$ be the random connectivity in $\mathrm{LP}_N$ induced by t…

In the setting of variational problems with a coercive conformally invariant Lagrangian of quadratic growth, consider its critical points. Hildebrand's regularity conjecture. Every…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

Fix $q\le 4$ and $p=p_c$. Let $(\Omega_\delta,a_\delta,b_\delta)$ be Dobrushin domains approximating a simply connected domain ${\bf \Omega}$ with marked boundary points ${\bf a}$…

A configuration is either a pair $(\ell,\Sigma)$ consisting of a Riemann surface $\Sigma$ and a loop $\ell\subset\Sigma$, or a triple $(\ell,\Sigma_1,\Sigma_2)$ consisting of Riema…

Self-avoiding-walk SLE conjecture. For $x=x_c$, the law of $\gamma_\delta$ in $(\Omega_\delta,a_\delta,b_\delta)$ converges as $\delta\to0$ to chordal Schramm–Loewner Evolution wit…

Let $\Omega$ be a simply connected open domain in the plane, and let $A,B,C,D$ be four boundary points of $\Omega$ in clockwise order. For critical percolation on the $1/n$-scaled…

Reversibility conjecture. After a time-change, $(W_0(\beta_0(1/t)))$, $0<t<\infty$, has the same distribution as $(\beta_0(t))$, $0<t<\infty$.

Let $D$ be a simply connected domain in the complex plane, and let $a,b,c,x$ be boundary points in counterclockwise order. Suppose that $D$ is conformally equivalent to the equilat…

Let $T^*$ be a triangulation of the torus, and let $\hat T^*$ be its universal cover. For $\alpha\in\mathbb{H}$, write $\hat T^*_{\alpha}$ for the corresponding realization in the…

Let $q\in[0,4]$, and consider the random cluster model on lattice domains. For the Ising model, let $F(z)$ denote the function in Proposition, which is discrete holomorphic and sat…

Let $q\in[0,4]$, and consider the Fortuin–Kasteleyn random-cluster model on the square lattice at the self-dual value $p_{\mathrm{sd}}=\sqrt q/(\sqrt q+1)$, with Dobrushin boundary…

Let $n\in[0,2]$ be the loop weight and $x>0$ the edge weight in the hexagonal-lattice $O(n)$ loop model, whose configuration weights are proportional to…