15 problems

Ergodicity conjecture. Under this condition, the system should be ergodic.

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…

For general $q$, let $f_1$ be the primitive arising from the first term in the integrand, and let $\Delta_{2q}(k)$ denote the corresponding function whose $(2q-1)$st derivative at…

Let $J=J(0,T)$ be the current, and consider equal reservoir densities $\rho_1=\rho_2=\rho=1/2$. Polynomial-cumulant conjecture. The cumulants of $J$ are simply polynomials in the a…

Consider the dynamic exclusion-process environment with particle jump rate $\nu>0$, and let $v(\rho)$ denote the speed of the random walk at density $\rho$. A zero-speed interval m…

Let $X$ be the random walk in the exclusion-process environment at density $\rho$, and let $v(\rho)$ denote its speed. In the symmetric case, meaning that the walk's transition pro…

Let $G$ be an arbitrary graph with arbitrary transition rates, let $n$ be its number of vertices, and let $\mathrm{EX}(k)$ denote the exclusion process with $k$ particles. Write…

Optimality conjecture. Theorem is optimal on any finite weighted graph, and more generally on resistance spaces; in particular, the preceding inequality should be an equality in th…

Let $\rho(t,u)$ denote the macroscopic density, with initial profile $\rho_0(u)$, for a system whose slow-site rate is $g(n)=\alpha n^{-\beta}$ with $\alpha>0$. At the critical exp…

Let $E_r$ be the $r$-th cumulant of the current, let $L$ be the system size, let $x$ be the asymmetry parameter, and let $\rho$ be the density. The intermediate regime is … It lies…

Let $h_r(\Phi)$ be the functions defined by … Here $B_k$ denotes the $k$-th Bernoulli number. Small-$\Phi$ asymptotics conjecture. For every $r\geq3$, … as $\Phi\to0$. This asympto…

Let $\xi$ be the discrete-time pure exclusion process, so that $\beta=0$, with asymmetry parameter $p$ and hybrid-zone size $|\xi_t|$ at time $t\in\mathbb{Z}^+$. The hybrid-zone gr…

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$ and state space $\mathcal{D}$. A process is recurrent from $S_0$ when…

The passage-time moment conjecture. Suppose that $p\leq1/2$ and $\beta\in[0,1]$. For any $\varepsilon>0$ and $S_0\in\mathcal{D}\setminus\{\mathcal{D}_0\}$,

Let $\xi$ be the exclusion-voter mixture process with parameters $\beta,p$, relaxation time … and state space $\mathcal{D}$, with distinguished state $\mathcal{D}_0$. The process i…