9 problems

The meandric-system scaling-limit conjecture. Uniform meandric systems converge in the scaling limit to the LQG sphere with central charge $\mathbf c_{\mathrm L}=28$, together with…

Let $(\mathcal M^\circ,P^\circ,\Gamma^\circ)$ be the infinite random planar map decorated by a bi-infinite Hamiltonian path and loops and possibly infinite paths associated with th…

Let $(\mathcal M,\Gamma)$ be the planar map and loop collection associated with the UIMS, and let $\ell_0$ be the loop or infinite path in $\Gamma$ passing through $0\in\mathbb Z$.…

Let $(\mathcal M,P,\Gamma)$ be the infinite random planar map decorated by a bi-infinite Hamiltonian path and the loops and possibly bi-infinite paths associated to the UIMS. Unifo…

Let $\mathcal L$ and $\mathcal R$ be independent two-sided simple random walks on $\mathbb Z$ with $\mathcal L_0=\mathcal R_0=0$, and use them to construct the two infinite non-cro…

Let $\mathfrak S_n$ be a uniform meandric system of size $n$, and order its loops by decreasing size. For each fixed $k\in\mathbb N$, let the $k$th largest loop be measured by its…

A meandric system of size $n$ is a collection of non-crossing loops intersecting the horizontal axis exactly at the points $0,\ldots,2n-1$, up to continuous deformation fixing the…

Let $n$ be a positive integer. A meandric partner of a partition $\pi\in\mathrm{NC}(n)$ is a partition $\rho\in\mathrm{NC}(n)$ such that the associated meandric system…

Linear-growth conjecture. The sequence $c_n'$ should follow a regime of “constant times $n$”; numerical experiments suggest that the constant is approximately $0.23$. This conjectu…