13 problems

Let $d$ and $n$ diverge with $n\leq 2^d$, let $G$ be a group of order $n$, and let $H=\mathbb{Z}_2^d$. For fixed $\varepsilon,\varepsilon'\in(0,1)$, write…

Cheeger-constant conjecture. There exists a constant $c$ such that

Let $G=\mathbb{Z}_n$ with $n$ prime, and fix the number of random generators $k$. Let $\operatorname{diam}(G_k)$ denote the diameter of the random Cayley graph. Diameter limit conj…

Let $q$ be prime, let $g_1,\ldots,g_k$ be independent uniformly random generators of the cyclic group ${\mathbb Z}_q$, and let $Diam(q,k)$ denote the diameter of the resulting dire…

Let $A$ be a set of size $s$ with doubling constant bounded by $K$, meaning $|A+A|\leq K|A|$. For each $ell$ let $\mathcal{F}_\ell$ be a collection of sets, and let $\tilde{O}$ and…

Product-condition conjecture. For random Cayley graphs of Abelian groups, the product condition implies cutoff. The product condition is known to be necessary for cutoff, but its s…

Lower-central-series comparison conjecture. If

The entropic-time conjecture. Under conditions similar to those in the paper's total-variation cutoff result, with high probability the random walk on $G_k$ exhibits cutoff in the…

Spectral-order conjecture. The correct order of $\Phi_*(G_k)$ is $\sqrt{\gamma}$, namely, using the stated order of the spectral gap, it is of order

Diameter concentration conjecture. Whenever

Typical-distance universality conjecture. The random variable $f(Z)$ concentrates at some value $\mathcal{D}$.

Diameter limit-law conjecture. The diameter divided by

Cheeger lower-bound conjecture. There exists an absolute constant $c$ such that, for all $\varepsilon\in(0,1)$, there are constants $N(\varepsilon)$ and $M(\varepsilon)$ such that,…