32 problems

Let $\rho_{\mathtt{FE}}$ be the critical density of the fixed-energy activated random walk on $\mathbb{Z}^d$, depending on the dimension $d$ and sleep rate $\lambda$, and let…

Positive critical regrowth conjecture. The critical regrowth parameter is positive:

Stabilizability critical-density conjecture. If $\rho<\rho_c$, then $\nu$ is stabilizable; if $\rho_c<\rho\leq 2d$, there exist stationary and ergodic measures $\nu$ that are not s…

Let $E_c$ be the energy threshold, $N$ the system size, and $d$ the dimension parameter. Let ${\cal T}$ be the dynamics, $\mu$ its invariant measure, and $P_i^k$ and $P_j^l$ distin…

Let $E_c$ be the energy threshold, $N$ the system size, ${\cal T}$ the dynamics, $\Omega$ an initial state, and ${\cal S}$ the singularity set. The distance between a point and a s…

In sandpile models, grains are added only when the dynamics eventually stops, as in the original Bak–Tang–Wiesenfeld model, and dissipation is localized at the boundaries. Vanishin…

Let $\Lambda$ be a $d$-dimensional square lattice of edge length $L$, with $N=L^d$ sites, and let $E_c>0$ be the critical energy. A configuration is stable when every site energy l…

Let $P_L(x)$ be the stationary probability distribution of an avalanche observable $x$ in a system of characteristic size $L$, and let $\tau_x$ be its power-law exponent. Critical-…

Levine–Liang mixing-time conjecture. The mixing time is

In driven-dissipative dynamics, particles are repeatedly added and the configuration is stabilized on a finite graph with a designated sink, while fixed-energy dynamics starts from…

Stochastic differential approximation conjecture. The process $\mathcal{X}_n$ behaves like a continuous-time process $X_t$ satisfying

Let $(\mathcal{X}_n,\mathcal{Z}_n)$ denote the process in the ballistic aggregation model, and let $\theta$ be its parameter. For $\varepsilon>0$, consider the regime in which…

Self-organized criticality density conjecture. The typical density $\rho_s$ in the steady state of the driven-dissipative system corresponds to $\rho_c$.

In the discrete Bak–Sneppen model on a circumference with $N\ge 3$ sites, each site $i\in[N]=\{1,2,\dots,N\}$ has a fitness value $\eta_i(t)\in[0,1]$ at time $t$, and at each step…

Stochastic-driven nonlinear dynamical systems (SDNDSs) are used as solvers for combinatorial optimization, with graph connectivity affecting the number of terms in the cost functio…

Let $\zeta_{\mathtt{DD}}$ be the limiting density of the driven-dissipative abelian sandpile model on increasingly large finite boxes, and let $\zeta_{\mathtt{FE}}$ be the critical…

Consider the Activated Random Walk model on a finite box of a vertex-transitive locally finite graph, in which particles are added one at a time at uniformly chosen sites, particle…

Let $S_n$ be the total spin under the law $b[0m\mu_n$ of the Ising-Curie-Weiss model, and let $r_n$ denote its interaction range. Assume that … The variance parameter from the fini…

Density conjecture. The critical density and the self-organized critical density should coincide:

The Zhang model has continuous non-negative height variables $z_i$, a common threshold height $z_i^c$, and random additions chosen from an interval $[a,b)$ with $0\leq a<b$. A site…

A stochastic differential equation has a stochastic evolution operator acting on differential forms over its phase space, and all such equations possess a topological supersymmetry…

Consider the email-communication model in which incoming and outgoing messages occur according to independent Poisson processes with intensities…

Consider the Bak-Sneppen model with finitely many species arranged on a circle, whose fitness values evolve by replacing the minimum-fitness species and its two neighbours with ind…

Forest-fire cluster-shape conjecture. In this forest-fire model, typical clusters are again subcritical Galton–Watson trees before gelation and critical Galton–Watson trees past ge…

Typical-cluster conjecture. Past gelation, typical clusters are again $cP(1)$ Galton–Watson trees.