8 problems

Let Assumption $\mathrm{(function)}$ hold, and set $q=q(N)=e^{-\gamma(N)/N}$, where $\gamma(N)\to+\infty$ and $\gamma(N)\ll N$. Let $\lambda^r_j(N)$ denote the parts of the random…

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 $M$,

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

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…

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…

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…

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…