174 problems

Approximation conjecture. There exists $\delta\in(0,\infty)$ such that, for every $\Delta\in(0,\delta)$, the process above is a stationary INARMA process, and the family of point p…

Let $u$ be the transition rate from the lounge to the queue, let $\beta$ satisfy … where $\eta>0$ is constant, and let $A=\lfloor\eta\mu/\alpha\rfloor$. For each $u$, write…

Let $(J_n)_{n\in\mathbb{N}}$ be a sequence of nonnegative random variables, and let $(\lambda_n)_{n\in\mathbb{N}}$ satisfy the limiting conditions denoted by … -- … -- … does not e…

For every stochastic transition matrix $M=(m_{ij})_{i,j\in\{1,2\}}$, every $\alpha_1,\alpha_2\in[0,1)$, and every $C_1,C_2>0$, the two-site moving-target search problem has an opti…

Convergence-rate conjecture. Theorem … and $$ increase with $\alpha$.

Let $\theta_n$ satisfy the scaling condition in the source, and let $X_n=(X_n(k);k=0,1,\dotsm)$ be the breadth-first walk on the random graph $\mathscr{G}_n^\theta$ constructed for…

Let $f$ be an interaction function satisfying the conditions of Theorem 2, and let $n$ be finite. A function … should exist such that $\{h(\Phi_t)\}_{t\in\mathbb{N}}$ is a non-nega…

Shift-invariance conjecture. If $\\{\alpha,\beta\\}$ is not an inversion of $w$, then the three probability ratios satisfy

Let $D\subset\mathbb R^d$ be the domain of a renormalization class $\mathcal W$, let $(c_k)_{k\geq0}$ be positive migration constants, and define $s_n=\sum_{k=0}^{n-1}c_k^{-1}$ and…

Let $z$ be a fluid limit with initial condition $z(0)=x$, where $x=\lim_n x^n/|x^n|$, and let $V_i$ denote the neighborhood associated with coordinate $i$. The right derivative at…

Let $v(x)$ be an antisymmetric velocity field with $v(x)>0$ for $x>0$, and suppose … Let $b(x)$ denote the diffusion-off boundary. Near-field scaling conjecture. Then … This predic…

Let $R(t)$ and $B(t)$ be the red and blue regions of the two-species competition model, and let $\mathcal{S}$ be the Richardson-model limit shape. For a subset…

Consider a balls-in-bins process $(I_m(i))_{i\in[B],m\geq 0}$ with feedback function $f$ in the almost-balanced regime. Here $[B]=\{1,\ldots,B\}$, and $\Pi$ ranges over all permuta…

Let $\kappa>4$ and set $\kappa'=16/\kappa$. In the configuration $(\mathbb{H},x,y,z,\infty)$ with $x<y<z$, let $(K_t)$ be the Loewner chain obtained by running an…

In the random-cluster (RC) model and the competing urns model, let $X_e$ denote the indicator associated with an edge or site, and let $S$ be any subset of the relevant index set.…

Graded-graph path-count conjecture. One has

Consider the random location deaths (RLD) process with birth probability $p$, death probability $q=1-p$, and parameter $r$. Let $L_n(a)$ be the number of species alive at time $n$…

Consider the random location deaths (RLD) process: species are born with probability $p$, die with probability $q=1-p$, and, on a death event, the least fit species is removed with…

Consider the annihilating version of the Pool model, in which excess particles at an attached location are annihilated, and take the particle density at criticality, $\lambda=1$. C…

Consider the Brownian Pool model obtained by replacing the continuous-time and continuous-space random walks in the Pool model by Brownian motion, and let $\lambda=1$ be the critic…

Let $\mathcal{E}_t$ denote the pool radius or size at time $t$ in the Pool model, and let $\lambda$ be the particle density. Critical Pool-model linear-growth conjecture. At the cr…

Let a rational arrival process (RAP) have joint densities on $(0,\infty)^k$ for every $k\ge 1$, and let $\bm{G}_0$ be the matrix in a RAP representation governing the evolution bet…

Matching conjecture. For all $a\geq 0$, the joint distributions of $(\hat{\mu}_a^{\infty}(i),\;i\geq 1)$ and $(\mu_a^{\infty}(i),\;i\geq 1)$ are equal. As a corollary, using the ex…

Diffusive-limit conjecture. As $\eta$ goes to zero, the processes $({\mathbf{X}}^\eta_t({\mathbf{x}}))_{t\ge 0}$ converge in law to the $2$-dimensional process…

A stochastic mass-action reaction system (SMART) is a continuous-time Markov chain describing molecule counts in a mass-action reaction network, and it is weakly reversible when ev…