14 problems

Let $d\geq 5$ and consider a discrete torus of side length $L$. For the Ising model and self-avoiding walk, let their two-point functions be defined in the usual way, and let…

Scaling-limit conjecture. As $\delta\to0$, the measures ${\bf P}_{D,\delta}$ converge to a probability measure ${\bf P}_D$ on simple paths from the origin to $\partial D$, with

A self-avoiding random walk is a lattice path that visits no lattice site more than once. Its scaling limit means the limiting continuum law obtained after rescaling the lattice sp…

Self-avoiding walk variation conjecture. We conjecture that

Four-dimensional self-avoiding walk conjecture. In the critical dimension $d=d_c=4$, SAW should still converge to Brownian motion when the scaling factor $C n^{-1/2}$ is replaced b…

Self-avoiding walk universality conjecture. SAW and SRW should have the same large-scale behaviour when $d>4$ but not when $d<4$.

Universal-profile conjecture. The profiles computed for self-avoiding walk and branched polymers on the complete graph should also apply to the corresponding SR and LR models at an…

Slade's conjecture. The susceptibility $\chi_N(z)$ for the hypercube remains of order $V^{1/2}$ throughout the critical window:

Let $z_L(\lambda)$ denote the pseudocritical point considered in the source for a $d$-dimensional high-dimensional torus, with parameter $\lambda$. Let the critical window mean the…

Four-dimensional SAW scaling-limit conjecture. The scaling limit of the SAW on $\mathbb Z^4$ should be given by Brownian motion, with an extra logarithmic correction in the scaling…

Scaling-limit conjecture. The scaling limit of self-avoiding walk is SLE$_{8/3}$.

Let $d\geqslant 4$ and let $p$ denote the edge-retention parameter for bond percolation on $\mathbb{Z}^d$. Write $\mu_d(p)$ for the quenched connective constant of the self-avoidin…

Let ${\cal S}=\{z\in{\mathbb H}:0<\operatorname{Im}(z)<1\}$ be the strip, and consider self-avoiding walks in a strip of height $y$, starting at the origin and ending on the upper…

Let ${\cal S}=\{z\in{\mathbb H}:0<\operatorname{Im}(z)<1\}$ be the strip, let ${\bf Q}^y$ denote the probability measure on self-avoiding walks in the strip of height $y$, and let…