200 problems

Nienhuis's critical-point conjecture. The point $x=x_c(n)$ is critical. This is the conjectured critical point for the loop $\mathrm{O}(n)$ model throughout $n\in[0,2]$. The source…

Pointwise upper-bound conjecture. The pointwise upper bound

Critical torus cluster-size conjecture. The largest cluster with periodic boundary conditions should have the same scaling as the largest cluster with zero boundary conditions, nam…

Let $q$ be the number of states in the Potts model, and let $(_+)$ denote the specified antiferromagnetic critical curve in the model's phase diagram. Baxter's conjecture. The curv…

Let $r,s>0$, let $x\in\partial B(0,r)$, define … and let $\mathcal C_r(0)$ be the set of vertices in $B(0,r)$ connected to the origin by open paths contained in $B(0,r)$. Small-bou…

Largest-cluster size conjecture. As $r\to\infty$,

Critical percolation absence conjecture. If $p_c<1$, then there are no infinite clusters at $p_c$.

Let $d<6$ and $b1=d/3$. Consider long-range percolation on $Z^d$ with parameter $b1$. Logarithmic-correction conjecture. The same logarithmic corrections computed for hierarchical…

Let lattice trees (LTs) and lattice animals (LAs) be considered on a lattice, and let $d_c$ denote their upper critical dimension, namely the dimension above which their critical e…

Triangle condition conjecture. One has $\nabla_{p_c}<\infty$.

Critical survival-probability conjecture. Under this condition,

Let $G$ be a Euclidean lattice, and let $\theta(p)$ denote the probability that the cluster of a fixed vertex is infinite at percolation parameter $p$. Critical percolation conject…

Critical-energy conjecture. The critical energy is exactly

Scattering conjecture for critical NLS. There exist functions

Let $W_{r,\lambda,z,w,A}$ be the measure on self-avoiding excursions in $A\subsetneq\mathbb{Z}^2$ from $z$ to $w$, assigning an excursion $\omega$ mass … where $m^*(A;\omega)$ is t…

Let $d\geq 2$, let $\phi_{p,q}$ be the free-boundary random-cluster measure on the cubic lattice $\mathbb{Z}^d$, and define the percolation probability and critical point by … … Le…

Let $\theta(p)$ denote the probability that the origin belongs to an infinite open cluster for Bernoulli percolation on $\mathbb Z^d$, and let $p_c$ be the critical probability. Cr…

Invasion percolation one-point function conjecture.

Aizenman's critical torus scaling conjecture. The largest cluster should have size of order

Borgs–Chayes–Kesten–Spencer conjecture. The scaling and hyperscaling hypotheses are valid whenever $n\leq 6$.

Critical-dimension conjecture. The critical dimension is

Critical free-energy power-law conjecture. The third derivative of the free energy satisfies

Free-energy singularity conjecture. There exists a constant $\alpha$ such that

Consider percolation at its critical parameter $p_c$, and let $Q(p)$ denote any of the quantities to which the conjectured scaling behavior applies. The source discusses the limit…

Let $k$ be the degree of a random $k$-regular graph, and consider bond percolation on it. Critical-component-size conjecture. At the threshold parameter … the largest critical comp…