33 problems

Fix $k\ge 2$, and for each $n$ let $K_n$ be the Burnside process on $(C_k^n,S_n)$. Consider starting states whose limiting empirical distribution on $C_k$ assigns a positive propor…

Let $(G_n)_{n\geq 1}$ be a sequence of finite rooted graphs, and let $t^n_{\textup{mix}}(\varepsilon)$ be the total-variation mixing time of the continuous-time simple random walk…

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…

Peres's cutoff conjecture. A process exhibits cutoff if and only if it satisfies the product condition

Consider the non-conservative symmetric simple exclusion process on a segment of size $N$ with reservoirs, and let $t_{\mathrm{mix}}(\epsilon)$ denote its mixing time for fixed…

Fix an integer $d$ and choose a $d$-regular graph uniformly at random from the graphs on $n$ vertices. The random walk is said to have cutoff when its distance from equilibrium dro…

Brownian motion and cutoff conjecture. For every $t\geq 0$, the state $\varphi_t$ is not absolutely continuous with respect to the Haar state. Moreover,

Multi-stack and restricted shuffle cutoff conjecture. Both chains exhibit cutoff, with the multi-stack chain around time

Ordered labeled cutoff-time conjecture. Both chains exhibit cutoff around time

Ordered labeled product-chain conjecture. The chains $\mathbf{X}^{\operatorname{ord,lab}}$ and $\mathbf{X}^{\operatorname{ord,lab}}_{\operatorname{prod}}$ exhibit cutoff around the…

$k$-ball cutoff conjecture. For $k=o(n)$, the balanced chain exhibits cutoff around the time

Let $(G_n)_{n\ge 1}$ be a sequence of finite vertex-transitive expander graphs, and consider simple random walk on each $G_n$. Levin–Peres conjecture. The sequence $(G_n)_{n\ge 1}$…

Let $(G_n)$ be a sequence of graphs with uniformly bounded degrees and diverging sizes. Let $(\varepsilon_n)$ be a sequence of constants in $(0,1)$, and let $(G_n^*)$ be as in the…

Let $(G_n)$ be a sequence of graphs with uniformly bounded degrees and diverging sizes, let $(\varepsilon_n)$ be a sequence of constants in $(0,1)$, and let $(G_n^*)$ be the graph…

Consider the symmetric simple exclusion process with reservoirs on a segment of size $N$, with parameters $p$ and $q$, and let $t^*$ denote the characteristic mixing-time scale. In…

Cutoff without the reservoir condition. Cutoff should still occur without being restricted by the condition $\frac{1}{1-p}=N^{o(1)}$. In particular, the conjecture asserts cutoff a…

Let $P$ be the transition matrix of a lazy random walk on $S_n$ generated by a conjugacy class or a union of conjugacy classes having $n-o(n)$ fixed points. Suppose that $P$ exhibi…

Consider the asymmetric simple exclusion process on a segment of size $N$ with open boundaries, in which particles have a drift, may move in both directions within the segment, and…

Let $X$ be a sequence of Ramanujan graphs, let $n$ denote the number of vertices, let $d$ be their common degree, and set $p=d-1$. Let $d_2(t)$ be the average squared $\ell^2$ dist…

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…

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…

Let $\operatorname{OST}_{n,\alpha}$ be the biased one-sided transposition shuffle with weight parameter $\alpha\in(0,\infty)$, and let $t_{n,\alpha}$ be the total-variation cutoff…

Let $\operatorname{OST}_{n,w}$ be the biased one-sided transposition shuffle, where $w(j)/j$ is monotonically decreasing, and let $N_w(n)$ and $N'_w(n)$ be the quantities used in t…

Let $\operatorname{OST}_{n,w}$ be the biased one-sided transposition shuffle, where $w(j)/j$ is monotonically decreasing, and let $N_w(n)$ denote its normalising quantity. Cutoff c…

Consider the simple exclusion process with open boundaries, with phase parameters $a$ and $b$, and let $t^N_{\mathrm{mix}}(\varepsilon)$ denote its $\varepsilon$-mixing time. Maxim…