141 problems

Let $\gamma\bin(0,2]$ and let $\eta$ be such that quantum natural time $\text{QLE}(\gamma^2,\eta)$ has been constructed. Let $\eta$-DBM denote diffusion-limited aggregation with pa…

Recurrence conjecture. The simple random walk on the $\nu$-IIPT is almost surely recurrent for every value of $\nu$.

Liouville quantum gravity scaling-limit conjecture. The limit in law as $a\to0$ of $( ua,nua^{partial})$ exists in the space of Radon measures equipped with the topology of weak co…

Let $h$ be a variant of the Gaussian free field on a planar domain, let $dx^2+dy^2$ be the Euclidean metric tensor, and let $\gamma\in(0,2)$. The associated Liouville quantum gravi…

Di Francesco–Golinelli–Guitter meander asymptotic conjecture. There exists a positive constant $R$ such that

Marckert–Mokkadem conjecture. The scaling limit of random quadrangulations should be given by the Brownian map.

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 ISE denote Integrated SuperBrownian Excursion, and let a Continuum Random Map (CRM) be a continuum analogue of the labelled-tree construction that encodes distances. For scaled…

Let ISE denote Integrated SuperBrownian Excursion, and let a Continuum Random Map (CRM) mean a continuum random metric-space object analogous to the Continuum Random Tree (CRT), wh…

Let $r_n$ be the radius of a random quadrangulation with $n$ faces. Schaeffer's radius scaling conjecture. There exists a constant $c$ such that … The paper states this conjecture…

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…

Consider uniform triangulations $T_{n,g_n,\mathbf{p}^{n}}$ in a regime where the total boundary length satisfies $|\mathbf{p}^{n}|\sim \alpha n$ for some $\alpha>0$. Local-limit co…

Scaling-limit conjecture. LQG surfaces should arise as scaling limits of random planar maps.

Let $D_\Phi$ be the $\gamma$-Liouville quantum gravity metric on a $\gamma$-LQG surface, and consider random planar maps equipped with their graph distance. LQG metric scaling-limi…

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…

Fix ${\mathbf{c}}\in(1,25)$ and let $Q=\sqrt{({\mathbf{c}}-1)/6}$. For each $k\in\mathbbm N$, let $(\mathcal M_k,\mathcal L_k)$ be the loop-decorated supercritical random planar ma…

Conformal embedding conjecture. Conformally embedded rooted series-parallel maps converge towards the $2$-LQG quantum sphere.

Universality conjecture. In view of the results for the model considered in the paper, the analogous results hold for any model of rigid critical $O(2)$ loop-decorated maps.