61 problems

Let $N$ be odd, and consider the stationary points emerging from the origin at the desynchronisation bifurcation $gamma=gamma_1$. The stationary points have an index, with index-…

Let $N$ be odd, and let $\gamma_1-\delta<\gamma<\gamma_1$ for the positive parameter $\delta=\delta(N)$ from the desynchronisation bifurcation theorem. Let $\cS(\gamma)$ be the set…

Stochastic Ising models are systems of spins with random dynamics exhibiting metastable behavior, namely long-lived states before a transition to a more stable state. Metastability…

Let $\bm{+1}$ and $\bm{-1}$ denote the all-positive and all-negative configurations, respectively. For a path $\rho$, let $\Phi_\rho$ be its maximal energy along the path and let…

Metastable consensus-time conjecture. If $p\geq p_c$, then there exists some $\bar c(p)>0$ such that

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

The finitary zero-one law is used to obtain bounds on metastable statements in probability theory. The authors give examples involving Kolmogorov's three-series theorem and bond pe…

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…

Consider the stochastic nearest-neighbor Kuramoto model with interaction range $r>1$, system size $n$, twisted states indexed by $q$, prefactor $C(q,n)$, and energy barrier…

Let $\mathcal{C}_N^\star$ denote the critical set associated with the metastable transition on the random graph with $N$ vertices, let $K_N^\star$ be the corresponding critical-set…

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…

Metastable systems exhibit behaviour that appears stable on some time scale but becomes unstable on longer time scales. In this setting, an inference scheme is metaconsistent when…

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,…

A complexity blowup is the increase in quantitative complexity incurred when passing to the relevant uniform or metastable formulation of a result. The conjecture concerns ordinary…

Consider a zero-range process on a fixed graph $G$ with $N>0$ particles and interaction parameter $alpha>1$, whose level-two large-deviation rate function is denoted by…

Let $L,C,R$ be the parameters of system … admits a unique stable limit cycle containing $\mathcal{S}$, $\mathcal{A}$, and $\mathcal{B}$ in its interior on the subdomain (iv)$_+$. T…

Three-time-scale conjecture. There exist precisely two more metastable time scales beyond $\theta_1$, namely $\theta_2$ and $\theta_3$, and all metastable transitions are observabl…

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 $v,-v\in S^1$…

Nonlinear-diffusion metastability conjecture. The same results should hold in the nonlinear diffusion case with $Q(0)=0$.

Caballero–Wayne conjecture. Perturbations by damping are formally related to perturbations by noise.

Let the sample space of a ferromagnetic Potts model be equipped with a Markov chain whose update rules are local, such as Glauber dynamics. Metastability conjecture. The performanc…

Conjectured vanishing and ordering. It is conjectured that

Let $\beta=\beta_L$ be an inverse-temperature sequence and let $\beta_L$ satisfy … Under this condition, the invariant measure is concentrated on the energetic valleys around the g…