18 problems

Let $n$ denote the size of a triangulation, and let the average number of its nodes of degree $d$ be divided by $n$, where $d$ is an integer larger than $2$. Polynomial degree-dist…

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…

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. The separatin…

Fix $\theta\in(0,1/2)$ and a sequence $(g_n)$ such that $g_n/n\rightarrow\theta$. Let $\mathbf{T}_{2n,g_n}$ be a uniform triangulation of genus $g_n$ with $2n$ faces. A simple sepa…

A simple separating non-contractible cycle (SNCC) is a simple cycle in a triangulation that is both separating and non-contractible. Let $t$ be an arbitrary triangulation without l…

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…

Uniform surface conjecture. For every fixed genus $g$, a.a.s.,

Let $\mathcal{S}^{(d)}_n$ be the set of $n$-vertex triangulations of the $d$-sphere, let $M(\mathcal{S}^{(d)}_n)$ denote their minimum volume, and let $\mu^{(d)}_n$ be the median o…

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…

Let $x_*$ be the phase-transition value separating the branched polymer phase, defined by $x>x_*$, from the crumpled phase, defined by $x<x_*$. Let the Continuum Random Tree (CRT)…

Let $\nu_u$ be the limiting transverse measure obtained from a convergent sequence of uniform probability measures on finite sphere triangulations, and let $G$ be the associated gr…

Let $T$ be a triangulation, let $x$ be a tangent vector on $T$, and let $G^x$ denote the corresponding graph of discrete tangent vectors. Suppose that $G^x$ is recurrent and that t…

Let $\mathcal{D}$ be trivial, and let $\nu_k$ be the uniform probability measure on sphere triangulations with at most $k$ faces. The measures are regarded as transverse measures,…

Let $\operatorname{Prob}_{p,q}$ be the law of a critical Ising triangulation of the disk with boundary parameters $p$ and $q$, and let $\operatorname{Prob}_{\infty}$ denote the inf…

Let $T_n$ be a uniform triangulation with $n$ faces, and perform a critical site or bond percolation on $T_n$. For random variables, write $X_n\approx n^{\alpha}$ when, for every…

Let $T^{(\ell)}$ be a critical random Boltzmann triangulation of the $\ell$-gon, with its simple boundary colored black, and let $\mathfrak{C}^{(\ell)}$ be the boundary cluster of…

Let $M$ be a surface of genus ${\bf g}$, let $\mathcal{T}_{N,\mathbf{g}}$ denote the set of triangulations of size $N$, and let $\nu_a$ be the random measure defined by … for posit…

Let $\mathcal{T}_{N}$ be the set of triangulations of the sphere with $N$ faces, and let $\mathcal{T}_{N,3}$ be the set of triangulations with $N$ faces and three marked faces. Equ…