33 problems

Universality conjecture. The geometry of large random maps should be universal: there should exist ensembles of random metric spaces depending on a small set of data, such as the c…

Let $g\geq 0$ and $p\geq 3$ be fixed integers, and let $\boldsymbol{\alpha}=(\alpha_1,\alpha_2,\ldots,\alpha_p)$ satisfy $\alpha_k>0$ and $\sum_{k=1}^p\alpha_k=1$. Define … For…

Fix a sequence $(g_n)_{n\geq0}$ such that … Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ on $3n$ edges, and let $\mathbf{T}_{2n,g_n}^*$ be its dual map. For…

A high genus map is a combinatorial map obtained by gluing polygons along their edges, with both its genus and size tending to infinity. An expander subgraph is a subgraph with uni…

Let $\bm{G}_{g,L}$ be the random trivalent unicellular metric map of genus $g$ with boundary length $L$, and let $\bm{T}$ be the random infinite trivalent metric tree with independ…

Let $x_n,y_n$ be two uniform, independent vertices of $\mathbf{T}_{2n,g_n}$. Let $m_{\theta}\in(0,1)$ be the parameter defined by the preceding relation between $h_{\theta}$ and…

GKN99 conjecture. The scaling limit of the problem of Hamiltonian cycles on random bicubic maps corresponds to two-dimensional quantum gravity coupled to a conformal field theory w…

Gaussian-fluctuation conjecture. The defect of the kernel satisfies

Let $\mathrm{M}_n(\mathrm{f}_n,\mathrm{g}_n)$ be random maps with face parameters satisfying $1\ll \mathrm{f}_n+2\mathrm{g}_n\ll n$. In the regime where…

Gromov--Hausdorff coupling conjecture. The variables $\mathbf U_{n_g,g}$ and $\mathbf S_g$ can be coupled so that

Unicellular-map approximation conjecture. The random hyperbolic surfaces $\mathbf S_g$ can in some sense be approximated by the rescaled unicellular maps $\mathbf U_{n,g}$.

Let $g_n$ satisfy … and let $\mathbf{T_{n,g_n}}$ be a uniform triangulation of genus $g_n$ with $3n$ edges. An induced subgraph is a $\kappa$-expander if every set of vertices has…

For any map $m$, let $\operatorname{SNC}(m)$ be the size of the shortest separating non-contractible cycle of $m$. Let $q^{(n)}$ be the random uniform high-genus quadrangulation, a…

Let $q^{(n)}$ be the random uniform high-genus quadrangulation, and let $\operatorname{syst}(q^{(n)})$ denote the size of its shortest non-contractible cycle. Systole tightness con…

Fix $\theta\in(0,1/2)$, let $g_n$ be the genus sequence used for the high-genus quadrangulations, and let $Q(n,g)$ denote the number of quadrangulations with $n$ faces and genus…

Let $q^{(n)}$ be the random uniform quadrangulation in the high-genus regime, with genus parameter determined by $\theta$, and let $\operatorname{diam}(q^{(n)})$ denote its graph d…

Let $(\mathbf{f}^{n})_{n \geq 1}$ be a sequence of face degree sequences, and let $(g_n)$ be a sequence such that $v(\mathbf{f}^{n},g_n)\geq 2$ for all $n\geq 1$. Assume that…

Let $\alpha=2$ and let $\nu$ be a critical equivalent of a weight sequence of $\alpha$-stable type, with $\nu$ having infinite variance. Let $(\tilde{B}_n)_{n\geq 0}$ be the sequen…

Let $S_n^O$ be the random surface in the BM model, and let $\widetilde{S}_n^O$ denote this surface conditioned on having a single vertex. Let $\widetilde{S}_n^C$ be the compactifie…

Compactness conjecture. Compactness, and consequently the existence of subsequential scaling limits, should still hold for essentially simple toroidal triangulations rooted at a un…

Universality conjecture. The sequence of rescaled metric spaces should converge and, up to a deterministic multiplicative constant applied to the distance, the limit should be the…

Let $(\mathcal{P}_{n})_{n \geqslant 1}$ be a good sequence of configurations. For each $n$, let $\mathbf{G}_{\mathcal{P}_{n}}$ be the associated random graph, let $\mathbb{G}_{n}$…

Consider random knot diagrams generated by the sampling method studied in the paper, and compare their asymptotic statistics with those of other random-knot models, such as self-av…

Let $\kappa_n(K)$ denote the number of random knot diagrams with $n$ crossings and fixed knot type $K$, let $N_K$ be the number of prime components of $K$, and let $\tau_0$ be the…

Let ${\bf X}$ be the self-similar growth-fragmentation with extinction time $\epsilon$, let $\alpha$ be its self-similarity parameter, and let $\omega_{+}$ denote the distinguished…