56 problems

Fix $d\ge 2$. Let $\eta=(\eta(x))_{x\in\mathbb{Z}^d}$ be an i.i.d. initial mass configuration for the divisible sandpile, with mean density $\rho$, and let…

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…

For every infinite, locally finite, transitive graph $G=(V,E)$, let $\mathbb{P}_p$ denote Bernoulli bond percolation with parameter $p$, let $o\in V$, and define…

Let $(X_i)_{i\ge1}$ be iid centered integrable random variables, let $(Y_i)_{i\ge1}$ be iid nonnegative random variables independent of $(X_i)$, and set $S_n=\sum_{i=1}^nY_i$. If,…

Let $X=(X_t)_{t\ge 0}$ and $Y=(Y_t)_{t\ge 0}$ be independent Markov processes taking values in a common metric measure space $(E,d,\mu)$. Define the collision-time and collision-po…

LLL mixing and density conjecture. (i) The sequence $(|\mu_{k(i)}|)_{i=1,2,\ldots}$ is strongly mixing as a stochastic process. (ii) Each $|\mu_{k(i)}|$ is contained in a compact s…

Laplace-transform criterion. The function $L_m$ is the Laplace transform of some random variable if and only if $m=0$, corresponding to a trivial random variable, or $m\ge e$.

Let $X$ be a second countable locally compact Abelian group, let $G$ be the subgroup of $X$ generated by all elements of $X$ of order $2$, and let $\alpha$ be a topological automor…

Let $X$ be a random variable, and let $n\geq 3$. Suppose the characterization theorem is formulated under condition. Non-necessity conjecture. For $n\geq 3$, condition is an artifa…

Let $Z=(Z_t)_{t\geq 0}$ be the stable process considered in the paper, let $T$ be the associated stopping time, and let $\alpha$ be its stability index. For every $k>0$, the moment…

Let $S_N$ be the sum of the discrete random variables considered in the system, and let $\mathbb{E}S_N$ denote its expectation. The coefficient matrix of is the matrix determining…

Let $\mathfrak{B}_0\supset \mathfrak{B}_1\supset\cdots$ be an $m$-dimensional linear-fragmentation process, and for $1\leq d\leq m$ let $\mathfrak{m}_d^{(N)}$ denote the maximum pr…

Let $F$ be a multivariate distribution admitting a density, with the relevant multivariate iteration map and a suitably reformulated version of Theorem 3. Multivariate extension co…

Let $F$ be a bivariate distribution admitting a density, and let the iteration map and the conclusion of Theorem 3 be understood as in the source paper. Extension conjecture. Theor…

Let $(\mathfrak{X},d,\mu)$ be a metric Polish probability space. Equivalence conjecture. The space $(\mathfrak{X},d,\mu)$ is asymptotically average-case approximable if and only if…

Let $(\mathfrak{X},d,\mu)$ be a metric probability space, where $\mathfrak{X}$ is a compact, connected, $D$-dimensional Riemannian manifold, $d$ is its geodesic distance, and $\mu$…

Let $\Lambda>1$ and let $\alpha\in(0,2)$. For each $\theta>0$, let $c_\theta$ denote the coefficient defined in Proposition corresponding to the asymptotics of the characteristic f…

Quantum law of large numbers. Under these assumptions, $e^{A_1t/n}\cdots e^{A_nt/n}$ converges in probability in the strong operator topology of $\ell_p$ to…

Let $\mathcal M_{z,z';\theta}$ be the boundary z-measure on the Thoma simplex $\Omega$, whose points are written $(\alpha;\beta)$ with nonincreasing nonnegative coordinate sequences…

A two-dimensional noise is a probability space $(\Omega,\mathcal F,\mathbb P)$ equipped with sub-$\sigma$-algebras associated with open rectangles in $\mathbb R^2$ and measure-pres…

Asymptotic logarithmic periodicity conjecture. The sequence $p_n$ does not converge as $n\to\infty$ over the positive integers, but it does along every subsequence $(n_j)$ for whic…

Let $B_x$ be the shifted unit ball at Euclidean distance $x$ from the origin, let $\tau_x$ denote the first passage time of the branching random walk to $B_x$, and let…

Nonzero-limit conjecture. The limit $L$ exists and is different from zero.

Density conjecture. The class of QID distributions on $\mathbb{R}^{d}$ with finite quasi-Lévy measure and zero Gaussian variance is dense in the space of probability distributions…

Let $f$ be the characteristic function of a probability measure with a nontrivial absolutely continuous component, and let $\varphi$ be the density function of this component. Let…