26 problems

Let $\mathcal{M}$ be the polling-system Markov chain, let $\ell$ be the boundary level, and let $\pi(x,y,s)$ denote its stationary probability at state $(x,y,s)$. In the spiral-spi…

Let $\mathbf{K}$ be a compact convex set of area $1$, let $\lambda>0$, and let $C^\star$ be the unique element of $\operatorname{argmax}\Phi_{\lambda,\mathbf{K}}$, where … Let…

Taillard's conjecture. For a fixed number of machines $M$,

Let $p\in(0,1]$, let $A_{p,n}$ be the total gelation time, and let $U_{p,n}$ denote the number of unicycle components at time $A_{p,n}$. Krapivsky's conjecture. … This conjecture c…

Let $L(\sigma)$ denote the statistic defined for the wreath product action, with $\sigma$ sampled from the relevant wreath product of parameter $k=2$. Wreath product longest increa…

Maximum-leaf-height constant. The constant in the asymptotic relation $D_n^* \sim c\log n$ in probability is

Common-descendant asymptotic conjecture. There exists a constant $\upsilon>0$ such that

Fix $k\in\{2,3,\ldots\}$ and sample $k$ vertices uniformly at random from the vertices of the random polytope $\operatorname{conv}\Pi_{d,1}$. Let…

Let $X(u)$ be a Gaussian random field, and let $X_{\mathrm{max}}=\max_{u\in M}X(u)$. Write $\mathbb{P}_{\mathrm{tube}}(X_{\mathrm{max}}>c)$ for the tube-method approximation to…

Kamčev–Liebenau–Wormald conjecture. There exists a set $\mathfrak{W}$ having probability $1-O(n^{-\omega(1)})$ in both $\mathcal{D}_r(n,m)$ and $\mathcal{T}_r(n,m)$ such that, unif…

Higher-order asymptotic conjecture. There exist appropriate constants $\gamma_2,\ldots,\gamma_{r-1}$ such that

Uncrossed-gap threshold conjecture. Uncrossed gaps appear at this scale; more precisely, under the scaling $R_L=\gamma\ln L/(\ln\ln L)^\theta$, the expected number of uncrossed gap…

Dominant-disconnection conjecture. Isolated nodes and uncrossed gaps together account for most of the probability that the graph is disconnected, with their contribution converging…

Let $b\geqslant2$ and consider the border aggregation model on a regular $b$-ary tree. Let $\xi_K$ denote the number of particles, or equivalently the corresponding growth time, ne…

Let $M_n$ be the finite honeycomb-lattice region, let $O$ be its central cell, and let $A_{k,n}\subset\Omega$ be the event that fluid $k$, for $k=1,2,3$, percolates from $O$ to the…

Asymptotic eigenvalue-dominance conjecture. This relation holds for arbitrary $m$ and arbitrary configurations of $\Sigma$ and $\Omega$.

Let $\mathcal{K}_n^{(d)}=\{K_n^{(d)}(t)\}_{0\le t\le1}$ be the $d$-Linial–Meshulam complex process, and let … Here $L_k(\mathcal{K}_n^{(d)})$ is the normalized $k$-th lifetime sum…

Rank conjecture. We conjecture that

Partial-trace remainder conjecture.

Let $R_n$ and $B_n$ denote the random reinforcement variables in the urn model, and suppose that the conclusions of Theorems 1–2 hold under the assumption $R_n=B_n$. Conjecture. Th…

Consider a chirping pattern consisting mostly of rises, with the position $n(k)$ of its $k$th fall and the distance $p(k)$ between the $k$th and $(k-1)$st falls. Let $P_n$ denote t…

Let $k$ be fixed, and let an alternating sign matrix (ASM) of size $n$ be chosen uniformly at random. Consider the numbers and positions of the entries $(-1)$ and $1$ in its first…

Asymptotic normality conjecture. The central string functions and the basic specialization of $\mathcal{F}_{\mu^{(N)}}$ behave asymptotically normally as $N\to\infty$.

Let $\theta,i,d\in\mathbb{N}$ with $i\leq d$, and let $p_c(i,d)$ be the threshold at which an open $i$-dimensional subgraph first appears. Let $\mathcal F_i$ denote the collection…

Let $\vec{\sigma}=(\sigma_1,\sigma_2,\ldots)$ be a sequence of nonnegative real numbers with mean $\alpha$, meaning that … Permutations of $[n]$ selected according to the weights…