15 problems

Let $p_a$, $p_b$, $p_c$, and $p_d$ be the edge-density quantities associated with the dimer model on an $n\times n$ torus in the thermodynamic limit. Consider an infinite sequence…

Consider perturbed simply attractive potentials, including anisotropic potentials, meaning potentials that are not necessarily isotropic. Let the variational principle be the relat…

Let $\mathcal{R}$ and $\mathcal{R}'$ be two surfaces in $\mathcal{S}$. Suppose the cell of $\mathcal{R}$ at position $(i,j)$ has cross ratio $k_{i,j}$, while the cell of…

Let $(X_k)_k$ be a sequence of random finite-area hyperbolic surfaces such that … Write $\lambda_1(X_k)$ for the first positive Laplace eigenvalue. Random hyperbolic-surface spectr…

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…

Large-genus Brownian-surface geodesic length spectrum conjecture. As $g\to\infty$, the random point process $c\,\Xi_g$ converges in distribution to $\mathcal{P}$:

Let $\mathbb{X}\subset\mathbb{S}^2$ be the embedded random surface, and let $\mathbb{X}'$ be an independent copy of $\mathbb{X}$. A neighborhood of $\mathbb{X}$ means a neighborhoo…

Let $\mu$ be a probability measure and let $[\mu]_\tau$ denote its free compression for $\tau\in[0,1]$. Define the sub-probability measure $\mu_\tau$ of total mass $\tau$ by…

Let $(\boldsymbol{\lambda}_n)_{n\ge 0}$ be the eigenvalue sequence of Liouville Brownian motion on a random surface $(D,h)$, where $D$ is the domain and $h$ is the Gaussian free fi…

Let $\mathbb{P}_{g,k(g)}$ be the probability measure on hyperbolic surfaces of genus $g$ with $k(g)$ cusps, and let $X \in \mathcal{M}_{g,k(g)}$ be a random surface. Assume that…

Let $X$ be a random compact hyperbolic surface of genus $g$, sampled according to the Weil–Petersson probability measure. Write $N_X^\Delta(0,a)$ for the number of eigenvalues of t…

Random-surface eigenfunction sup-norm conjecture. With probability tending to $1$ as $g\to+\infty$,

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…

Let $U:\mathbb{R}\to\mathbb{R}$ be smooth and satisfy $U(x)=U(-x)$ and … for some $\alpha>0$. For each $M>0$, define the interaction sequence $u^M=(u^M_n)$ by $u^M_n:=U(n/M)$. For…

Let $N=1,2,\dots$, let $\varkappa_j=(\nu_j,\eta_j,\tau_j)\in\mathcal D$ be distinct triples, and let $\Omega_j=\Omega(\nu_j,\eta_j,\tau_j)$. The correlation formula of Theorem 1.17…