43 problems

Let $B_r$ be the ball of radius $r$ around the root in the type II uniform infinite planar triangulation, and let $\overline{B}_r$ be its hull, obtained by adjoining to $B_r$ all f…

Let $M\in[6,\infty]$, and let $T^M_j$ be chosen randomly uniformly among isomorphism classes of spherical triangulations with $j$ vertices and maximum degree at most $M$. Given…

Weak-limit conjecture. The unique maximal Schnyder wood of the UIHPT is the weak limit of the maximal Schnyder wood of a uniformly random triangulation from $\cT_n^m$ as…

For each $n\in{\bf N}$, let $M_n$ be a random planar map sampled as in the planar-map model above, and embed it into $\mathbbm S^2$ or a subset of $\mathbbm C$ by a discrete confor…

Let $M_n$ be a random planar map with $n$ edges whose law is given, for each fixed map $\mathfrak m$ with $n$ edges, by … where $\Delta_{\mathfrak m}$ is the graph Laplacian and…

Let $Q\in(0,2)$ and let $p_c(Q)\in(0,1)$ be the critical probability from the non-trivial percolation transition conjecture. Consider critical site percolation on $\mathcal M_k$ wi…

Fix $Q\in(0,2)$ and consider site percolation on $\mathcal M_k$ with open boundary condition: boundary vertices are open, while each interior vertex is independently open with prob…

Let $\mathcal M_k$ be the supercritical random planar map, and let its free uniform spanning forest (FUSF) be the forest obtained from the free boundary condition. FUSF connectedne…

Let $Q=\sqrt{({\mathbf{c}}-1)/6}$ and let $X^k$ be the continuous-time random walk on $\mathcal M_k$ reflected off infinity, stopped when it first hits the boundary at time…

Let $\mathcal M_k$ be the supercritical random planar map and let $H_k$ be its Tutte embedding into $\overline{\mathbbm D}$. For an end $\omega$ of $\mathcal M_k$, define its image…

Let $\{M_n\}_{n\geq1}$ be a sequence of infinite random planar maps believed to lie in the universality class of $\gamma$-LQG, such as uniform infinite triangulations when…

Let $q\text{-FK}$ planar maps denote the critical Fortuin–Kasteleyn planar maps with parameter $q$, and let $\gamma$ be a parameter in $[\sqrt 2,2)$ satisfying … Scaling-limit conjec…

The spanning-tree and GFF scaling-limit conjecture. Under an appropriate scaling as $n\to\infty$, the 4-tuple $(M,P,\phi_1,\phi_2)$ converges in law to a unit boundary length…

Fix $s>1$, and let $(M_k',L_k')$ be sampled from the modified iteration procedure. Let $\mathbf c_{\mathrm L}\in(1,25)$ be the central charge parameter determined by $s$ through ……

Fix $s>1$. In the modified iteration procedure, each hole of perimeter $m$ is filled using a ring whose inner boundary has length either $\lfloor sm\rfloor$ or…

For $k\in\mathbb N$, let $(M_k,L_k)$ be sampled from the probability measure on triangulations of the disk decorated by fully packed $O(2)$ loop configurations, at the critical par…

For $\ell>0$, let $\overline{\mathrm{MD}_{0,2}^n}(\cdot,[\ell n^{1/2}])^\#$ and $\underline{\mathrm{MD}_{0,1}^n}([\ell n^{1/2}])^\#$ be the probability measures on the two types of…

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…

Unmarked planar-map scaling-limit conjecture. There exist unique constants $d(\kappa)>0$ and $c,c'>0$ such that the rescaled spaces converge in law to…

Let $\widehat{q}(y)$ be the quantity defined in the paper from the harmonic measure of the embedded graph, and let $y$ tend to infinity in the Euclidean embedding. Harmonic-measure…

Let $Q_n$ be a uniform quadrangulation with $n$ vertices, let $Q_n^\dagger$ be its dual, and let $\tau_{Q_n}(\varepsilon)$, $\tau_{Q_n^\dagger}(\varepsilon)$ denote uniform mixing…