108 problems

Strauss's conjecture. The critical exponent is $\gamma(n-1)$. This conjecture was later confirmed with the complete high-dimensional theory.

Let $n$ be the spatial dimension, let $\mu>0$ and $\alpha>1$, and consider the small-data Cauchy problem … with $u_1\in L^1(\mathbb{R}^n)$. Let $\gamma$ be the function determined…

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, with $N\geq3$, and let $p>1$, $\lambda>0$. Consider the Neumann problem … where $\nu$ is the outward unit normal to…

Let $n\geq 4$, $\mu>0$, and consider the regular Cauchy problems … with small initial data, or … with small initial data. Denote by $p_S$ the Strauss exponent and by $p_F$ the Fuji…

Sobolev hyperbola conjecture. The Sobolev hyperbola is the critical dividing curve between existence and nonexistence of solutions to the Hardy–Hénon system. This conjecture propos…

Let $L$ be the $d$-dimensional cubic lattice, let $\omega$ be chosen uniformly from the $n$-step self-avoiding walks on $L$, and define the mean square displacement by … For $d=2$…

Let $L$ be the $d$-dimensional cubic lattice, let $c(n)$ be the number of $n$-step self-avoiding walks on $L$ beginning at the origin, and let … be the connective constant. For…

Let $\hat{S}$ be the loop-erased random walk in two dimensions, and let … Define the growth exponent by ${\bf E}[\hat{\sigma}_n]\approx n^\alpha$, or, when the limit is not known t…

Let $L$ be a lattice in ${\mathbb R}^2$, and let $\beta$ and $\gamma$ denote its critical exponents. Planar critical-exponent universality conjecture. The critical exponents of all…

For each $\lambda\geq -1/2$, let $r_\lambda$ be a critical parameter such that the total mass $|W_{r_\lambda,\lambda,0,N,\mathcal{H}}|$ neither decays nor grows exponentially as…

Extremality conjecture. In the class of operators defined by the ellipticity condition in, all critical exponents lie in the same interval; equivalently, the operators…

Let a loop-erased random walk in three dimensions be run until it reaches distance $r$, and let its number of steps be the quantity under discussion. Growth exponent conjecture. Ph…

Planar-lattice universality conjecture. The same asymptotic should hold for critical site percolation on any planar lattice.

The scaling limits of many two-dimensional systems in statistical physics are expected to exhibit conformally invariant behaviour at criticality. Conformal invariance conjecture. A…

Large-$k$ critical-exponent conjecture. The exponent $\nu$ tends to $1$ as $k$ tends to infinity.

Dotsenko's conjecture. The energy operator $\epsilon$ should correspond to the primary field $\phi_{21}$ for all $q\leq4$, implying

CAM hypothesis. The extrapolated zero crossings approach the critical point according to the envelope determined by the extrapolation, namely the negative slope at the crossing is…

Consider the two-dimensional random-bond Potts model in the large-$q$ limit, and let $\nu$ denote its correlation-length exponent. The random transverse-field Ising chain (RTFIC) h…

Let $p_c$ be the critical threshold of a hierarchical lattice. Distinguish the lattice's local structure from its global characteristics, such as its macroscopic dimension. Localit…

Let $H_n$ denote the maximal height of a geodesic from $0$ to $ne_1$, where $e_1$ is the first vector of the canonical basis of $\mathbb{R}^d$, and let $\xi(d)$ be a constant depen…

Critical-power conjecture. For $\alpha=0$, the small-data Cauchy problem admits the critical power

Ashkin–Teller magnetisation exponent conjecture. The exponent of the magnetisation fields is constant and equal to

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…

Equality of critical exponents. The equality $b=\beta$ should hold in every dimension larger than two. Equivalently, near the critical density, $a(\rho)\simeq (2^d-1)\rho_a(\rho)$.…

Let $d<6$ and consider the critical FK-Ising model on $\mathbb Z^d$. The four-point function refers to the four-point connectivity quantity controlled by the correlation inequaliti…