23 problems

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 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

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 $G$ be a group, let $k$ satisfy $k \gg \log |G|$ and $\log k \ll \log |G|$, and let $\mathcal{G}_k$ be the Cayley graph of $G$ generated by $k$ independently and uniformly chos…

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…

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…

Mixing-time and cutoff conjecture. The following claims should hold:

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…

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 $d$ 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…

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…

Symmetric cutoff conjecture. The lower bound in the symmetric pre-cutoff estimate is sharp, and cutoff occurs.

Let $\{G_n\}$ be a sequence of transitive graphs, and consider the Glauber dynamics for the Ising model on these graphs. Let $\beta_c$ denote the critical temperature, let $\delta$…

Let $(G_n)$ be any family of finite vertex-transitive expander graphs, and let the simple random walk (SRW) be the walk that at each step moves uniformly to a neighboring vertex. V…

Let $G\sim\mathcal{G}(n,3)$ be a random $3$-regular graph, and consider the lazy random walk on $G$. Its mixing time is measured in total variation and the lazy walk stays in place…