7 problems

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…

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 $J>0$ 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…

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…

Infinite Edwards-model conjecture. Under $\mathbb{Q}^{\beta}$, the process $X$ is transient and