67 problems

Consider the parking process on $\mathbb{Z}^d$ in which each site initially contains a car with probability $p$ and a parking spot with probability $1-p$, independently, and cars p…

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

Exponential-relaxation conjecture. If there exists $\kappa>0$ such that

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

Let $G=(V,E)$ be a finite graph with positive edge rates $r_{xy}$, and let $L$ be the generator of the symmetric simple exclusion process on the $n$-particle configuration space…

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…

Long-term behaviour conjecture. 1) If $\alpha<0$ and $\alpha+\beta\nu(G)<0$, then the process $X$ is positive recurrent with the stationary distribution $\mu$ defined in Theorem. 2…

Let $d\geq 2$. For $i=1,2$, let $A_i$ be the event that infection type $i$ reaches sites arbitrarily far from the origin, and let $A=A_1\cap A_2$ be the coexistence event. Let…

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…

Let ${\bf X}^0(t)$ be the East process on the positive integers, started from a single particle at site $0$. Let $r({\bf x})$ be the rightmost occupied site in configuration…

Small-voter-fraction nonergodicity conjecture. For every $p<1/2$, there exists a number $\beta_0(p)>0$ such that, whenever $\beta<\beta_0(p)$, the hybrid process $\tilde\eta_t$ wit…

Consider a finite multi-floor TASEP with $c\ge 1$ floors and attempted particle arrival and departure rates $\alpha>0$ and $\beta>0$. Let … and let … Assume condition (a): $L^*$ co…

Convergence conjecture. Under these assumptions,

PRC for simple IPS. Every simple IPS with positive rates is ergodic.

Let $\rho_t$ and $\mu_t$ denote, respectively, the laws at time $t \geq 0$ of solutions to the local-field equation (LFE) and the Markov local-field equation (MLFE). Let $d$ be a s…

Let an impurity in the totally asymmetric simple exclusion process have jumping rates $b1$ and $b2$, and initially be located at the interface of a $1$-$0$ profile. Assume that…

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…

Convergence conjecture. For $\alpha>\frac12$, the half-line open TASEP with step initial condition $\eta_t^{\mathrm{step}}$ converges to the probability measure…

Let an additive interacting particle system (IPS) evolve on a one-dimensional lattice with homogeneous, one-sided, nearest-neighbor interaction and positive rates. Additive ergodic…

Let $C_N$ be a cycle, let $K\ge3$, and consider the $K$-type competing urn scheme with a nonzero initial configuration. For each type $j$, let balls of type $j$ have independent Po…

Let $K\ge3$, let the underlying graph be either a hypercube or a discrete torus, and let the initial configuration be any nonzero configuration of the $K$-type competing urn scheme…

Let $N+1$ particles form a lattice model with elastic collisions. Define … so that $U_k$ is the speed of the centre of mass of the leftmost $k$ particles, while $U_{N+1}$ is the sp…

Jara's conjecture. The mixing time is of order