21 problems

Let $\widehat{\operatorname{Pol}}(n;1)$ be the space of confined equilateral $n$-gons, let $\phi_i$ be the $i$th turning angle, and let $\kappa$ denote total curvature. The limitin…

Let $d_k$ denote the $k$th chord length of a random confined equilateral polygon, and let $E_j$ denote the Euler number appearing in the expected chord-length formulas. For an inte…

Let $L$ be the curvature length scale and $\ell_0$ the lattice edge length for regular or semi-regular hyperbolic tessellations, and let $n$ be the polygon length. Hyperbolic-latti…

Let $\nu$ and $\vartheta$ be the critical exponents governing the continuum limit and the zero-cosmological-constant partition function of the SOP model. SOP critical-exponent conj…

Let SOP denote the model of self-overlapping polygons, with its multiplicity-weighted measure. SOP random-path conjecture. The SOP model fits into the general random-path framework…

Let a self-overlapping polygon have boundary length $2n$, and let its multiplicity be the number of distinct distorted disks it bounds. Exponential multiplicity conjecture. There e…

Let $w_{2n}$ be the number of self-overlapping polygons of length $2n$ in the SOP model, counted with multiplicity factor $\mu_\gamma$. SOP connective-constant conjecture. The limi…

Self-avoiding polygon Airy conjecture. The derivative of the scaling function is

SOP scaling conjecture. The same large-$n$ scaling is valid for self-overlapping polygons (SOP), modulo possible logarithmic corrections. This extends a scaling form observed in se…

Let $W_{2n}(t)$ be the generating function of self-overlapping polygons of boundary length $2n$, weighted by their area through $t$. The model has a critical point at $t=1$. Contin…

A linear polymer in dilute solution is sufficiently long and undergoes a ring-closure reaction, producing a ring polymer. Frisch–Delbrück conjecture. Such ring polymers should be k…

In dimension $3$, let $K^\text{Arm}(n,w,k,r)$ be the probability that a length-$k$ arc of a random open arm is an $r$-local knot, and let $K^\text{Pol}(n,w,k,r)$ be the correspondi…

Let $\operatorname{Arm}(n,d,w)$ be the space of open $n$-edge arms in $\mathbb{R}^d$ with fixed edgelength $w$, and let $\operatorname{Pol}(n,d,w)$ be the corresponding space of cl…

Let $\operatorname{Arm}(n,d,w)$ be the space of open $n$-edge arms in $\mathbb{R}^d$ with fixed edgelength $w$, and let $\operatorname{Pol}(n,d,w)$ be the corresponding space of cl…

Let a uniform cyclic quadrilateral have angles ${\Greekmath 010B}$, ${\Greekmath 010C}$, and ${\Greekmath 0121}$. The cyclic quadrilateral joint-density conjecture. The joint densi…

Let a uniform cyclic $6$-gon have consecutive polygonal angles $reekmath 010B]$, $reekmath 010C]$, $reekmath 010D]$, $reekmath 010E]$, and let ${\Greekmath 0119}$ denote the releva…

Let $H$ be a compact convex set with nonempty interior in the plane, and let $Z_0,\ldots,Z_{n-1}$ be chosen independently and uniformly at random from $H$. Write $Q_H^n$ for the pr…

Let $L_n$ be the random polygon sequence generated by the consecutive random subdivision, and suppose that $u$ satisfies … and that the support of $\xi$ contains at least two dist…

Trapezoidal change-region conjecture. In the even-vertex case, the typical change regions are trapezoids, which would imply that the speed of the process is

Even-vertex speed conjecture. The speed of the process is

Let $n$ be the number of segments in one of the closed random polygons considered in the paper, and let the total curvature be the sum of the turning angles between consecutive edg…