51 problems

Let $v_N$ denote the asymptotic velocity of the $N$-BBM. Brunet–Derrida velocity conjecture. As $N\rightarrow\infty$, … This conjecture predicts the leading finite-population corre…

Conjectured improved rate. The rate in the preceding estimate should be improvable to

Let $\psi:[0,1]\to[0,\infty)$ satisfy $\psi(x)>\psi$ for $x\in[0,\epsilon]$ for some $\epsilon>0$ and $\psi(1-h)=o(h)$ as $h\to0$. Asymptotic-velocity conjecture. The $(\psi,1,N)$-…

Lu–Lu–Nolen scaling conjecture. If the kernel is scaled proportionally to the number of particles as $N^\beta K(N^\beta\,\cdot)$, then the behavior of the particle system should re…

Crystallization conjecture. Interacting particles spontaneously arrange in periodic structures at low temperatures.

Let $\mu_t^\infty$ be a solution of the limiting dynamics whose initial condition is a finite mixture of spherical heat kernels, and suppose that one component collapses to a Dirac…

Barrier-height convergence conjecture. Under a suitable rescaling, the barrier height satisfies that $\xi_t/\sqrt{t}$ converges to some distribution.

Let $\mathcal{A}(M,P,N)$ be the single-level estimator, and let $\bm X_N^P$ denote the particle-system realization. For $M$ samples and $P$ particles, variance-decay conjecture. Th…

Let $\bm X^P$ be the continuous-time particle-system approximation and let $Z(t)$ be the true solution of the corresponding McKean–Vlasov equation. Let…

Let $g:\mathbb{R}\to\mathbb{R}$ be smooth with bounded derivatives, let $G^P(x)=\frac{1}{P}\sum_{j=1}^P g(x_j)$, and let $\bm X_N^P$ be a realization of the particle system with co…

Consider the open ASEP with light particles at the triple point $(A,C)=(1,1)$, and let $r=1$. Write $\mathbf{loc}_1$ for the location of the single light particle in a system of si…

Let $A$, $C$, and $r$ be the parameters of the open ASEP with light particles, and let $AC\leq 1$ be the fan-region condition in Theorem. The stationary measure of the standard ope…

Consider the Fleming–Viot particle systems $(\mathbf X^r_t)_{t}$ with empirical measures $\pi(\mathbf X^r_t)$, in the regime where both $n_r$ and $\underline\lambda_r$ tend to infi…

Speed-selection conjecture. Under mild conditions, if $Q$ is transient or null recurrent, the leftmost particle still has speed $v_0$; if $Q$ is positive recurrent, the speed $v$ s…

Let $N\geq 1$, let $\tilde{X}_N$ be the random number of particles remaining in the physical perfectly plastic collision process, and let $\tilde{Z}_N$ be the corresponding random…

Let $N\geq 1$ be the initial number of particles, and let $\tilde{X}_N$ denote the random number of particles remaining after the perfectly plastic collision process. A random vari…

Tableau-process weight conjecture. This weight is proportional to the weight derived from the tableau process of Ayyer et al.

Let $\boldsymbol{x}_{i}^{(t)}$ evolve according to the finite-particle system defined by the Gaussian-Stein dynamics above, and let the corresponding mean-field limit be the associ…

Let $F$ be an $(m-1)$-dimensional manifold with known chart $\bar g\in\mathcal{C}^2(V,F)$, where $V=[0,1]^{m-1}$ or $V=\Omega$, and let $T(y,F)$ be the tangential space at $y\in F$…

Unbounded-rate extension conjecture. The results of the main theorem should hold in situations where the branching rate is unbounded and thus where the Moran type process is undefi…

Roberts–Schweinsberg conjecture. The number of particles near $z_n$ in the long run should be proportional to

The particle system is considered with its initial condition as in Theorem 1.4 of the paper. Initial-condition perturbation conjecture. The perturbation in the initial condition of…

Let a particle system have a general injection distribution, possibly injecting particles to the right of the free boundary, and let its hydrodynamic limit describe the macroscopic…

Consider a competing particle system with $n\geq 3$ particles, and let $X(\cdot)$ denote its particle process. A solution is strong when it is adapted to the driving noise, and pat…

In the symmetric case, let $\lambda>0$ and $\mathrm u$ denote the parameters of the system, let $G(\cdot)$ and $H(\cdot)$ be the two gaps, and let $\bm\pi$ be the stationary distri…