20 problems

Metastable random-walk conjecture. As $\epsilon\to0$, the rescaled process approaches a symmetric simple random walk on

Let $\mu$ be a probability distribution on the non-negative integers satisfying … Let $(\mathscr{G}_n)_{n\in\mathbb{N}}$ be configuration graphs on $n$ vertices derived from $\mu$,…

Let $X$ be sampled from the ERGM measure $\mu$ on graphs with vertex set $[n]$, let $e$ be any edge in $\binom{[n]}{2}$, and let $p^*$ denote the corresponding Erdős–Rényi edge pro…

Let $\Gamma_N^\star$ denote the metastable energy barrier for the dynamics on the random graph with $N$ vertices, and let $\mathbb{P}_N$ be the corresponding probability law. Conje…

Harmonic-map metastability conjecture. The relaxation time of the corresponding Langevin dynamics grows exponentially fast with $\beta$ if and only if $\mathcal E$ has non-trivial,…

Let $I_n(t)$ denote the expected number of infected vertices at time $t$ for the contact process on the dynamical graph with initially all vertices infected. Write…

Let ${\mathcal U}$ be a critical two-dimensional update family. Its difficulty is … where a critical update family is balanced if there are no two opposite directions $v,-v\in S^1$…

Let $S_{\star}$ be the set of sites on which the condensate may reside, and let $r(x,y)$ be the jump rates of the underlying random walk. Let $Y^{\mathrm{rv}}(\cdot)$ be the Markov…

Let $S$ be the finite state space of the underlying random walk, with jump rates $r(x,y)$, and let $S_{\star}$ be the set of sites on which the condensate may reside. Define the Ma…

Metastable random-walk conjecture. (i) The metastable time satisfies $\tau^{(N)}=C(J_N,\sigma_N)e^{J_N/\sigma_N^2-\log N}$, where $C(J,\sigma)$ is polynomial in $J$ and $\sigma$. (…

Logarithmic extinction-rate conjecture.

Quasi-stationary extinction-event conjecture.

Quasi-stationary distribution conjecture. Then:

Generalized extinction conjecture. With a suitable modification of condition $$ , the conclusions of Theorem should hold for a general positive symmetric matrix ${\bf A}$.

Let $\mathcal{C}_n^\star$ denote the critical set of configurations, let $K_n^\star$ denote the associated quantity governing the transition from the critical set, and let…

Let $Gamman^\star$ denote the energy barrier and let $\mathbb{P}n$ be the probability law of the random graph. Scaling conjecture for the critical height. There exists a…

The discrete nonlinear Klein–Gordon equation is … Its quasi-stationary solutions have the form $u(t)=\cos(\mu t+\nu)\alpha_\mu$, where $\mu$ and $\nu$ are parameters and…

Three-dimensional threshold conjecture. There is a constant $C_{a,b,c}$ such that

Asymptotic constant conjecture. If $a<b$, then there is a constant $C_{a,b}$ such that

Lower-threshold conjecture. For any $\epsilon>0$, if