14 problems

Let $G_n$ be an expander sequence converging locally to the $d$-regular tree. Consider the local limits associated with item c., a sequence of $d$-regular expanders with $n$ vertic…

Consider the undirected preferential attachment model with colors, and seek its local limit as a rooted random tree. A local-limit conjecture is that a potential candidate is a bra…

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…

Let $\tau^{*}$ be a sub-critical and non-generic marked Galton–Watson tree with offspring distribution $\mathbf{p}$ satisfying the condition in … . Let $M(\tau^*)$ denote the total…

Let $G_n$ denote the sparse exponential random graph model under consideration, and let $a$ be the parameter appearing in the asymptotic number of triangles. Consider also the mode…

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…

Let $(\mathcal{G}^{N,p^N,\kappa})$ be a sequence of $k$-type frozen percolation processes approximating a type flow $pi$, and let $G(N\pi(t),\kappa(t))$ denote the corresponding in…

Let $\mathcal{C}(t)$ be the steady state cluster growth process, and let the mean field forest fire model of Ráth and Tóth evolve in its stationary state. Consider the cluster cont…

Infinite-boundary limit conjecture. There exists a distribution $\widehat{\mathbb{P}}^{(\infty)}_{\mathsf{q}}$ on one-ended maps with an infinite simple boundary such that

Let $\mathcal{T}_{n,g}$ be the set of rooted triangulations of genus $g$ with $n$ vertices, and let $T_{n,g}$ be uniformly distributed in $\mathcal{T}_{n,g}$. For…

Fix $\kappa\in(0,\frac{2}{27}]$, and let $\mathbf{T}_{\kappa}$ be the corresponding infinite triangulation. Let $T_{n,g}$ be a uniformly rooted triangulation of genus $g$ with $n$…

For $g\geq 0$, let $\mathcal{T}_{n,g}$ be the set of rooted triangulations of the torus of genus $g$ with $n$ vertices, and let $T_{n,g}$ be uniformly distributed in…

The paper constructs a family of random infinite triangulations, and let $T_{n,g}$ denote a uniformly chosen rooted triangulation of genus $g$ with $n$ vertices. Local-limit conjec…

A local limit of the planar maps $M_n$ is an infinite graph whose finite neighbourhoods of faces around the root edge have limiting probabilities as $n\to\infty$. The infinite unif…