22 problems

For every infinite, connected, vertex-transitive, cubic graph $G$, let $c_n(G,v)$ denote the number of self-avoiding walks of length $n$ starting at a vertex $v$, and let the conne…

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…

Suppose $G=(V,E)$ is a finite, connected, simple graph. For $x,y\in V$, let $\mathscr{S}(x,y)$ be the set of self-avoiding walks from $x$ to $y$, and let $\mathscr{S}(u_i,v_j)$ den…

Let $\mu$ be the connective constant, let $\operatorname{SAW}_k$ be the set of self-avoiding paths of length $k$, and write $\eta\text{ escapes }\zeta$ for the event that the infin…

Let $d\geq1$, and let $\operatorname{SAW}_\infty$ be the space of infinite self-avoiding walks. For a walk $\eta$, write $\eta[i,\infty)$ for its tail and $\eta[0,k]=\zeta'$ for it…

The critical-exponent asymptotic conjecture. For every $d\ge2$, such a critical exponent exists, and

Consider continuous $n$-step self-avoiding walks on the hyperbolic space $\mathbb{H}^d$, with steps of length $\varepsilon$ and with the distance between every pair of vertices con…

Radial SLE scaling-limit conjecture. There exist $b$, $\tilde b_n$, a critical value $\beta=\beta_{\cent}$, and a partition function $\Psi^*(D;{\bf z},0)$ such that, as…

Universal-cover connective-constant conjecture. There exists $k\in\mathbb N\cup\{0\}$ for which

Self-avoiding-walk Wilfian conjecture. The sequence $a_n$ has no Wilfian formula of type (W1).

Convergence-diagram conjecture. The following convergence diagram holds:

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

Let $\mathcal{C}$ be an infinite subclass of self-avoiding walks in an upper half-plane, with starting point on the surface. For $\omega\in\mathcal{C}$, let $h(\omega)$ be the maxi…

Let $W(z;1,0;a)$ be the generating function for prudent loops, with step variable $z$ and surface fugacity $a>0$. Let $z_{1}^{\ell}\approx 0.412095$ denote a root of … and let…

Continuity conjecture. The connective constant is continuous with respect to local convergence:

Pivot time-scale conjecture. The integrated autocorrelation time of this observable is of the same order as the time required to make successful pivots at all length scales with re…

Let the $O(n)$ loop model be defined on the semi-infinite honeycomb lattice with the boundary oriented as in Figure(b). For $n\in[-2,2]$, assign fugacity … to occupied vertices and…

Let $\Omega\subsetneq\mathbb{C}$ be a simply connected domain, let $z\in\Omega$, and let $a,b$ be distinct boundary points, with the boundary smooth near $b$. For $\delta>0$, let…

Let $\Omega$ be a simply connected domain in $\mathbb{C}$ with boundary points $a$ and $b$, and let $(\Omega_\delta,a_\delta,b_\delta)$ be discrete approximations. At the critical…

Boundary and bulk lambda-self-avoiding-walk scaling conjecture. The partition function and rescaled measures satisfy

Self-avoiding-walk SLE conjecture. For $x=x_c$, the law of $\gamma_\delta$ in $(\Omega_\delta,a_\delta,b_\delta)$ converges as $\delta\to0$ to chordal Schramm–Loewner Evolution wit…