15 problems

For a planar lattice $\Lambda$ in Euclidean space, let $K_\Lambda$ denote the lattice-dependent constant governing the asymptotic critical-temperature scaling under iterative trian…

Self-avoiding loop scaling-limit conjecture.

Let $G$ be a planar lattice, let $p_c=p_c(G)$ be the critical point for regular site percolation, and define … where $\theta(p,\delta)$ is the probability that the origin belongs 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…

Consider the self-destructive percolation model on the square lattice or another planar lattice: first perform ordinary site percolation with parameter $p$, then make every site in…

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

Consider planar lattices in Euclidean space with maximal coordination number $q_{\rm max}$, and let the Apollonian lattices be the family obtained by iterative triangulation of the…

Let $T_{\rm c}^*(q_{\rm max})$ be a function of the maximal coordination number $q_{\rm max}$, defined as a continuous extension of the critical temperatures of the Apollonian latt…

Finite-reinforcement percolation conjecture. Assume that $d=k=2$. Then there exists $\alpha_0>0$ such that

Percolation conjecture for $d=k=2$. Assume that $d=k=2$. Then

Consider the reinforced $k$-out percolation model on the hypercubic lattice $\mathbb Z^d=(V,E)$, with edge weights initially equal to $1$. At each time step, every vertex selects a…

Order-of-transition conjecture. Among all random-cluster models defined on planar lattices, the phase transition is of first order if and only if $q>4$.

Planar-lattice equivalence conjecture. The conditions (a), (b), and (c) given in Theorem hold equivalently for every planar lattice $L$.

Let $x \in \mathbb{Z}^2$ be a neighbor of the origin, and let $p_N$ be the probability that $x$ lies on the loop-erased random walk from $o$ to $\partial G_N$. Loop-erased random w…

Planar universality conjecture. The critical exponents of all lattices in $\mathbb{R}^2$ are the same, with