43 problems

Scattering conjecture for critical NLS. There exist functions

Critical-dimension conjecture. The critical dimension is

Let $m(0)$ denote the odometer at the origin for fixed-energy activated random walk with i.i.d. initial particle density $\rho$, and let $\rho_c(\lambda)$ be its critical density f…

Non-concavity conjecture. There exists $\delta=\delta(k,d)>0$ such that $p\mapsto q_c(p)$ is not concave on $[0,\delta]$.

Pseudocritical-profile conjecture. For $d>d_{\mathrm{c},\alpha}$, there is a pseudocritical point

Table-profile conjecture. The profile functions $f(s)$ are the functions listed in Table for both SR and LR models in every dimension $d\ge d_{\mathrm{c},\alpha}$, including the up…

Universal-profile conjecture. The preceding susceptibility and two-point-function asymptotics hold for every listed model, with the same profile function for its SR and LR versions…

Let $d\geq 3$, and let $(\beta_{L^N})_{N\geq 2}$ denote the critical coupling constants associated with the $N$-th moments of the stochastic heat equation. Strict-ordering conjectu…

Square-root critical behavior conjecture. The formal matrix model has the asymptotic expansion

Critical percolation complexity conjecture. As $m\to\infty$,

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…

Let $W^*$ be the Brownian excursion of length $1$, let $\Psi(x)=\mathbb{E}\exp\left[x\int_0^1W^*(t)\,\mathrm dt\right]$ be the moment-generating function of its area, and define ……

Infinite-order criticality conjecture. The free energy satisfies

Critical-moment growth conjecture. The critical moment satisfies

Let $T_n$ be the triangle count under the edge-triangle model with law ${\mathbb P}_{n;\alpha,h}$, and let $(\alpha_c,h_c)$ be the critical parameter pair. At this pair, let…

Let $d\geq 1$, and let $J:\mathbb{Z}^d\times\mathbb{Z}^d\to[0,\infty)$ be a symmetric, integrable, translation-invariant kernel. Suppose that there are constants $0<\alpha<d$ and…

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…

Critical polynomial decay conjecture. One has

Critical extinction conjecture. For every finite nonempty $C$ and every admissible background configuration $B$, one has

Let $\mathcal U$ be a critical $d$-dimensional family, meaning that its stable directions satisfy the criticality condition, and let $L_c(\mathcal U,p)$ be the critical length at i…

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…

Critical symplectic-group conjecture. For these parameters,

Critical orthogonal-group conjecture. For every $k>1$ with $s=1/\sqrt{k}$,

Let $P_e(\delta)$ denote the optimal reconstruction error probability in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write…

Let $I(\delta)$ denote the mutual information in the binary-tree broadcasting model, let $\delta_c$ be the critical channel parameter, and write $\delta=\delta_c-\tau$ with…