124 problems

In two dimensions, let self-avoiding walk be rescaled in the usual way, and let $\operatorname{SLE}_{8/3}$ denote Schramm–Loewner evolution with parameter $8/3$. Lawler–Schramm–Wer…

Nienhuis's connective-constant conjecture. The connective constant of the hexagonal lattice equals

A self-avoiding walk (SAW) is a contiguous sequence of moves on a lattice that does not cross itself and does not visit the same point more than once. In two dimensions, consider t…

Critical-walk recurrence conjecture. The biased random walk $RW_{\lambda_c}$ on $T_{\mathbb H}$ or $T_{\mathbb Z^2}$ is recurrent.

Half-plane self-avoiding-walk measure conjecture. The measure satisfies

Consider a self-avoiding walk of length $n$ on the lattice, and let its endpoint displacement from the origin be measured by the mean distance. Flory's conjecture. The mean displac…

Let $c_n$ be the number of self-avoiding walks of length exactly $n$ in the square lattice $4\mathbb{Z}^2$, let $\mu_c$ be the connective constant defined by … and let $A$ be a pos…

Let $D\subset\mathbb{C}$ be a bounded domain with smooth boundary, let $z,w\in\partial D$ be distinct, and let the lattice spacing be $\epsilon$. At the critical parameter…

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 $b$ and $C$ be as in the conjectured asymptotic … After multiplying the union over boundary endpoints by $N^{2b-2}C^{-1}$ and scaling paths by $N^{-1}$, consider excursions in…

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…

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…

Self-avoiding walks are lattice paths that visit no site more than once. The Schramm–Loewner evolution with parameter $\kappa=8/3$, denoted $\operatorname{SLE}_{8/3}$, is a random…

Let $\operatorname{SLE}_{8/3}$ denote Schramm–Loewner evolution with parameter $\kappa=8/3$, and let self-avoiding walks be lattice paths that do not visit the same vertex more tha…

Let $\gamma$ be a very long rescaled self-avoiding walk in the half-plane, let $H$ be a suitable subdomain, and let $\Phi_H$ be the conformal map used in the half-plane restriction…

Let $\gamma$ denote a rescaled self-avoiding curve, and let $SLE_{8/3}$ denote chordal Schramm–Loewner Evolution with parameter $8/3$. Self-avoiding-walk scaling-limit conjecture.…

Let $D$ be a domain with boundary points $A$ and $B$, and let $P_{D,A,B}^{SAW}$ denote a scaling limit of measures on self-avoiding curves from $A$ to $B$ in $D$, assigning weight…

Self-avoiding-walk SLE conjecture. If the scaling limit of self-avoiding walk exists and is conformally invariant, then its only possible limit is $SLE_{8/3}$; the source further r…

Self-avoiding-walk dimension conjecture. It is conjectured that the scaling limit of planar self-avoiding walks has paths of dimension $4/3$.

Let the Brownian frontier denote the outer boundary of a planar Brownian curve, and let the scaling limit of long self-avoiding curves be the conjectural continuum limit of self-av…

Consider the uniform measure on self-avoiding walks of length $n$ in the upper half-plane, started at the origin, and its limiting law as $n\to\infty$. The half-plane self-avoiding…

Let an infinite self-avoiding walk be obtained as the limit of the uniform measures on self-avoiding walks of length $n$ in the upper half-plane, started at the origin, as…

SLE$_{8/3}$ scaling-limit conjecture. SLE$_{8/3}$ is the scaling limit of the half-plane self-avoiding walk.

The half-plane self-avoiding walk is expected to have a scaling limit described by the chordal SLE curve with parameter $\bkappa=8/3$. Scaling-limit conjecture. The SLE$_{8/3}$ cur…