17 problems

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…

Let $\chi^-_\nu$ be the exponent governing the large-area tail of the negative-curvature JT area distribution. Exact negative-curvature exponent conjecture. … This agrees with the…

Let $\nu$ and $\vartheta$ be the critical exponents governing the continuum limit and the zero-cosmological-constant partition function of the SOP model. SOP critical-exponent conj…

Let $p_c$ be the critical edge parameter and let $h_c(p)$ denote the Kertész line in the planar Ising case. The planar Ising Kertész-line conjecture. As $p\to p_c$, … for some cons…

The critical-exponent asymptotic conjecture. For every $d\ge2$, such a critical exponent exists, and

Let $m$ be a positive integer, let $\mathbf{H}_{\mathbb{C}}^{m}$ be complex hyperbolic $m$-space, and let $\Gamma$ be a convex cocompact isometry group of…

Let $B_R$ be the ball of radius $R$ in $\mathbb{R}^N$, let $N>k\geq 2$, and let $u\in C_c^k(B_R)$. Critical logarithmic Hardy conjecture. The inequality … holds. Moreover, the cons…

Consider an anisotropic bond percolation process on ${\mathbb {Z}}^d\times{\mathbb {Z}}^s$, with $d,s\geq1$, parameters $(p,q)$, and $p_c(d)-p>0$ sufficiently small. Let…

Let $q_c:[0,1]\to[0,1]$ be the critical function for anisotropic bond percolation, and let $p_c(d)$ denote the critical parameter for $d$-dimensional percolation. Write…

Let $G=\operatorname{Aut}(\mathbf B^n)$, and let $\Gamma<G$ be a convex-cocompact torsion-free subgroup. Let $\delta_\Gamma$ denote its critical exponent. Chengbo Yue's gap conject…

Let the Rudin–Shapiro sequence be viewed as an infinite word, and consider each nonempty factor as a circular word. Its circular critical exponent is the largest exponent of a repe…

Consider the critical three-dimensional Ising model, with spin variables $\mathit{\sigma_x}$ and critical Gibbs measure $\mathit{\mu_{\beta_c}}$. Three-dimensional Ising exponent c…

Consider a bond percolation model on $\mathbb{Z}^d\times\mathbb{Z}^{s}$ with parameters $(p,q)$, where $p<p_c(d)$. Let $q_c(p)$ be the critical value of $q$, and let $\chi_d(p)$ de…

Let $G$ be an isoradial graph with the bounded-angles property, let $k\in\mathbb{N}$, and let $\sigma\in\\{0,1\\}^k$. For a translated annulus $\mathcal{A}^u(N,n)$ centred at $u$,…

Let $\Gamma'$ be a decoration of a $d$-regular tree, and let $g(x)$ be the generating function for weighted clusters containing the origin. Let $D$ denote the exponential growth fa…

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. The cell-wise first-wave critical expone…

Let $\Gamma$ be an infinite $d$-regular tree, and let $\Gamma'$ be a graph quasi-isometric to $\Gamma$, including a decoration of $\Gamma$. A critical exponent is the power-law exp…