33 problems

Let $G$ be a finite group, let $\mathscr{M}_G$ denote the space of admissible functions on $G$ up to rescaling, and let $\Delta_G\subset\mathscr{M}_G$ be the locus of lattice system…

Faceted-droplet hypothesis. As $l\to\infty$, with probability tending to $1$, there exist six distinct two-dimensional planes $L_i=L_i(\sigma)\subset\mathbb{T}^3$, $i=1,\ldots,6$,…

Let $u(k,q)$ denote the size of the smallest set of uniqueness for the positive function class $\left(\mathcal{B}^{k}_{q}\right)_+$. In particular, consider the case $q=2$. Minimum…

Fix $k\in\mathbb{N}$. For the Ising partition function $Z_{\mathbb{Z}_m^t}(\lambda,\beta)$ on the even torus $\mathbb{Z}_m^t$, let $L_j$ denote the sum of the weights of clusters o…

Few-monochromatic-edges conjecture. For fixed $i,j\geq 0$, there exists a rational function $Q_{i,j}(g)$ in $g$ such that

Many-monochromatic-edges conjecture. For fixed $i,j\geq 0$, there exists a rational function $R_{i,j}(g)$ in $g$ such that

Let $\lambda\neq 0$ be fixed and let $(\omega_n,\sigma_n)$ be an FK-Ising coupling of an Ising model at $\beta=\beta_c+\frac{\lambda}{n}$ in $[-1,1]^2\cap\frac{1}{n}\mathbb{Z}^2$,…

Let $\chi(\beta)$ denote the susceptibility of the high-dimensional Ising model and let $\beta_c$ be its critical inverse temperature. Logarithmic susceptibility conjecture. Let…

Free-boundary effective critical point conjecture. There exists such a $\beta_{c,r}$ for which all statements of the non-Gaussian limit and universal profile conjecture hold with…

Non-gaussian limit and universal profile conjecture. As $r\to\infty$, $h_r^{-1}S_r$ converges in distribution under $\langle\cdot\rangle_{\beta_c+sw_r}^{\mathbb{T}_r}$ to…

Let $\mathcal G_\Delta$ be the class of graphs under consideration, let $\hat{\mu}_{G,\beta,\eta}$ be the fixed-magnetization Ising measure on $G\in\mathcal G_\Delta$, and let…

Scaling-limit factorization conjecture. There exists $\lambda_0=\lambda_0(U)>0$ such that, for all $|\lambda|\leq\lambda_0$, there exists $\beta^*=\beta^*(\lambda)$ for which, for…

Fix a nonzero parameter $\lambda$, and let $Z^{\sigma_\Lambda}_{G}(\beta)$ denote the partition function of the Ising model with boundary condition $\sigma_\Lambda$. Suppose it is…

Consider the two-dimensional Ising model and let $h_c(p)$ denote the Kertész line as a function of the parameter $p$, with $p_c=p_c(2,0)$ the zero-field critical point. Kertész-lin…

Gravitational-field variational-shape conjecture. The preceding convergence in probability holds. The supplied context says that the solution of this variational problem is known i…

Let $s\in\mathbb{S}^{d-1}$, let $n$ tend to infinity, let $\beta_{\rm sat}^{*}(s)$ be the saturation threshold for the truncated two-point function, let $\beta_{\rm c}$ be the crit…

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…

Let a similar finite-range, ferromagnetic, translation-invariant and reflection-invariant system be obtained by replacing the Ising spins in Theorem 1 with spin variables in the Si…

Kawasaki dynamics mixing conjecture. For $\beta<\beta_c(\Delta)$, the Kawasaki dynamics mix in time polynomial in $n$ for any fixed magnetization and any graph $G$ of maximum degre…

Let $\Delta$ be an integer with $\Delta \ge 3$ and let $\beta\in \mathbb{C}_{\mathbb{A}}$ with $\beta \not \in \mathbb{R}\cup \{i, -i\}$. Let $G$ be a graph with maximum degree at…

Near-critical scaling-limit conjecture. As $a\downarrow0$,

Let $G$ be an infinite Cayley graph. The magnetization $m^+(\beta,h)$ is the plus-state magnetization, with $\beta,h\in[0,\infty)$ and $\beta_c$ the critical inverse temperature. Di…

Let $(\mathfrak{t},\sigma)$ be sampled under $\mathbb{P}_{p,q}$, and let $\eta_{p,q}$ be the length of its leftmost interface. Let $E$ be the expectation under $\mathbb{P}_\infty$…

Let $\mathcal U$ be a finite family defining two-dimensional $\mathcal U$-Ising dynamics. A finite set $\mathcal T$ of stable directions determines a $(\mathcal T,L)$-droplet when…

Let $\mathcal U$ be a critical two-dimensional family, meaning that its stable directions intersect every open semicircle and have finite intersection with some semicircle. Start t…