242 problems

Maximal-current mixing-time conjecture. Theorem 1 remains valid in the maximal current phase for general $A,C\leq 1$.

Let $rho0: mathbb{R}rightarrow[0,1]$ be an initial density profile, and let ${nu^{epsilon}}{epsiloninmathbb{R}{0}}$ be probability measures on the configuration space…

Ferromagnetic ordering conjecture. The spectral gap $\gamma_k$ is independent of $k$; equivalently,

The delayed Poissonian mean-field model is an interacting-particle system with Poisson-like attributes, and its corresponding mean-field PDE is intended to describe the large-parti…

In two-type first passage percolation on the lattice, two infection types spread simultaneously using passage times with potentially different intensities. The event that both infe…

Consider a two-type SIS model in which the two species have general, possibly unequal, death rates and compete for space. For each species, define its birth/death ratio as its birt…

Positive Rates Conjecture. Every IPS on a one-dimensional lattice with homogeneous interactions of bounded range and positive rates is ergodic.

Deffuant phase-transition conjecture. There is a phase transition from discordance to consensus at the confidence threshold $\theta=1$.

Stationary-distribution conjecture. For each $\theta>1$, the process $Z_t$ admits exactly two stationary distributions: the trivial one, the Dirac mass at $0$, which is reached onl…

The mixing time is the time required for a Markov particle system to approach its stationary distribution. A particle system is in the maximal current phase when its stationary par…

Let $I_N$ be the lattice site set, let $\Omega_N$ be the configuration space, and let $\{\mu^N\}_{N\in\mathbb N}$ be probability measures on $\Omega_N$ satisfying…

Poisson Hypothesis. Spiking deliveries to distinct replicas shall be asymptotically distributed as independent Poisson (or compound) point processes.

Let $D(\rho)$ denote the diffusion coefficient of the Kob–Andersen model at density $\rho$, with parameters $k$ and $d$. In the case $k=d=2$, the high-density regime concerns…

Equality of critical values conjecture.

Consider majority dynamics on $\mathbb{Z}^2$ with i.i.d. initial opinions of density $p\in[0,1]$, and let $\eta_t(0)$ denote the opinion at the origin at time $t$. Exponential fixa…

Let the symmetric Neuhauser–Pacala model be the competing-species system on the lattice $\mathbb Z^d$, with coexistence meaning that both species persist in the long term. Neuhause…

Consider the two-type Richardson model with infection rates $\lambda_1$ and $\lambda_2$, and let $A$ denote the event that both types grow indefinitely from their finite initial co…

Complete convergence conjecture. Complete convergence should hold more generally when $a=a^\dagger$ and $\Lambda$ is amenable, but not in general on nonamenable lattices. Complete…

Let $D\subset\mathbb R^d$ be open, bounded, and convex, and let $\mathcal W$ be a renormalization class on $\overline D$. Assume that the asymptotic fixed-point equation has a uniq…

Consider two systems on the same lattice dimension $d$ with the same number of types $n$, having parameter vectors $(p_1,p_2,\ldots,p_n)$ and…

Let $n$ and $d$ be arbitrary, and let $p_1,p_2,\ldots,p_n$ denote the type-dependent poisoning parameters of the particle system. Starting from the all-zero configuration, suppose…

Consider the northeast model on the integer lattice, with all sites in the first quadrant initially set to spin $0$ and all other sites initially assigned independently by $p$-coin…

Consider a zero-range process on $L$ vertices with inhomogeneous rates … where $\lambda_x\in[a_1,a_2]$ for constants $0<a_1<a_2<\infty$, and where $c(\cdot)$ satisfies (H.1) and (H…

Let $\beta\in(0,\frac12)$ parameterize an ${\cal O}(n^{-\beta})$ perturbation of a constant equilibrium profile in the hydrodynamic limit, with time rescaled by $n^{1+\beta}$ and s…

Consider the multitype contact process with interaction range $r$ in dimension $d$, infection parameters $lambda1$ and $lambda2$, and frozen-state parameter $gamma$. Let $r0$ an…