179 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…

Let $X$ be a nonnegative generalized gamma convolution (GGC) random variable. Then, for every real $q\ge 1$, the random variable $X^q$ is also a GGC random variable. Equivalently,…

Let $d\ge 1$, let $E_d$ be the set of unoriented nearest-neighbor edges of $\mathbb{Z}^d$, and let $\omega=(\omega_e)_{e\in E_d}\in(0,\infty)^{E_d}$ be a translation-invariant ergo…

Let $\mathcal{C}_{\alpha}$ denote the $\alpha$-Cauchy distribution for $1<\alpha\leq 2$. Determine exactly for which values of $\alpha$ the law $\mathcal{C}_{\alpha}$ is self-decom…

For every admissible finite weighted graph or hypergraph update structure and every system size $N\ge 2$, the spectral gap of the corresponding conservative Kipnis–Marchioro–Presut…

For every integer $d\ge 2$, consider the nearest-neighbor contact process on $\mathbb{Z}^d$ with infection rate $\lambda$ and recovery rate $1$. Let $\lambda_1$ be the critical val…

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…

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…

Product-form characterization conjecture. The network $(\mathcal{G},\kappa)$ has a product-form stationary distribution if and only if it is complex balanced.

Continuity conjecture. The map

Suppose that $f:\mathbb{R}\rightarrow\mathbb{R}$ is an odd function of compact support such that $xf(x)\ge 0$. Let $X$ be the process defined by the Durrett–Rogers equation. Durret…

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 $z$ 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 $n$…