31 problems

Large-scale convergence conjecture. Under the scaling assumption on the noise, the expected density should satisfy

Let $\bm{u}(x)$ denote the polymer and let $\langle *\rangle$ denote its Gibbs measure. For points $x,y$ with large separation, consider the disorder-averaged squared displacement…

Consider a two-dimensional weakly self-avoiding star polymer with $N$ branches, polymer length parameter $T$, self-avoidance parameter $\beta$, and radius $R$. Daoud–Cotton star-po…

Let $R_{d,\beta,T}=(E^{Q_T}[B_T^2])^{1/2}$ be the radius of weakly self-avoiding Brownian motion in dimension $d$, with penalization parameter $\beta>0$. Universal scaling exponent…

A self-avoiding random walk in two dimensions is expected to have an end-to-end distance of order $T^{3/4}$ after $T$ steps. Self-avoiding walk scaling conjecture. The end-to-end d…

Simplified-model local limit conjecture. For all sufficiently large $n$, for the upper extremal point $M_n^+$ and every integer $k$,

Bessel coupling conjecture. There exists a coupling of $(R,Y)$ and $\widetilde{\mathbf{B}}$ such that almost surely there are $\delta_1>0$ and $n_1\in\mathbb{N}$ for which, for eve…

Local limit conjecture. Almost surely,

Let $\mathbb T=\mathbb T_{M_1,M_2}$ be a lattice tube, and let $L$ be a non-split link embeddable in $\mathbb T$. Let $p_{\mathbb T,n}(L)$ denote the number of $n$-edge embeddings…

For a knot type $K$, let $p_n(K)$ be the number of $n$-edge polygons of knot type $K$. Let $0_1$ be the unknot, let $f_K$ be the number of prime knot factors in the knot decomposit…

Let $K$ be a knot type, and let $p_n(K)$ denote the number of $n$-edge lattice polygons of knot type $K$. Let $0_1$ denote the unknot, and let $f_K$ be the number of prime knot fac…

Let $Q_L$ be the partition function of the interacting partially directed self-avoiding walk (IPDSAW) of length $L$, and suppose that in the collapsed phase it has asymptotic form…

Let $J>0$ be the intrinsic length of a moving polymer taking values in $\mathbf{R}^2$. Because the path $x\mapsto\mathbf{u}(t,x)$ behaves like a two-dimensional Brownian motion, it…

Consider the generalized Poland–Scheraga model with disorder generated from an interaction function $f$, and let … be the resulting disorder field. In the paper's construction, $f$…

Let the generalized Poland–Scheraga model have loop exponent $\alpha>1$ and Gaussian i.i.d. disorder. Smoothing exponent conjecture. The model should undergo a smoothing phenomenon…

Let ${\mathrm{MaxDisjtPoly}}_n(I,J)$ denote the maximum number of disjoint polymers with endpoints in intervals $I$ and $J$, and suppose the theorem in the source gives a rarity bo…

Boundary-range shape theorem conjecture. A shape theorem takes place on the scale

Let $\overline{\Psi} \in (0,\infty)^3$ satisfy $\Psi_2 \geq 1$. For $a \leq b$, let $\mathcal{C}_{0,*}([a,b],\mathbb{R})$ be the space of continuous functions…

Let $r$ denote the walk thickness, and let the scaling exponent be the exponent governing the growth of the walk's size with its length. Scaling-exponent convergence conjecture. Fo…

Consider an effective ensemble of $k$ ordered Ising polymer paths whose interactions have exponential decay with rate parameter $\chi>0$, competing with entropic repulsion between…

Let $d\geq 1$ and consider the interacting weakly self-avoiding walk of length $L$, with self-intersection parameter $\gamma\geq 0$ and self-touching parameter $\beta$. At critical…

Two-dimensional self-avoiding walks with generic short-range attraction undergo a collapse transition. The theta point is the tricritical point separating the dilute-polymer and de…

Localized-phase splitting conjecture. The localized phase splits into two subphases

Odd-even melting-temperature conjecture. The melting temperature computed by an extended dimer film model should still depend on whether the polymer length is odd or even.

Let $p(\beta,\nu)$ be the quenched free energy, let $\lambda(\beta)$ be the corresponding annealed free-energy function, let ${\cal D}(\nu)$ be the domain of admissible inverse tem…