13 problems

Poisson–Dirichlet conjecture. If $0<\alpha=\alpha_t<1$ is the macroscopic-cycle fraction, then

Let $G$ be a Cayley graph, let $(\psi_e)_{e\in E(G)}$ be independent unit-rate Poisson processes on the edges, and let $\beta>0$. At time $\beta$, swapping the particles across eac…

Tóth's macroscopic-cycle conjecture. For $d\geq3$ and $\beta>\beta_c$, macroscopic cycles form: roughly $\alpha L^d$ particles belong to macroscopic cycles and roughly…

Tóth's conjecture. The quantity

Let $G$ be a finite weighted hypergraph with non-negative weights. At each step, choose a hyper-edge with probability proportional to its weight and then choose a uniformly random…

Short-cycle conjecture. The permutation has only short and finite cycles.

Interchange cutoff conjecture. (i) Cutoff holds if and only if the product condition holds. (ii) If

Let $G$ be a finite connected graph. Let $\chi_G^{\mathrm{IP}}$ denote the comparison quantity introduced in the paper for the Interchange Process, and let…

Let $G=(V,E)$ be a graph, let $(V)_k$ denote the ordered $k$-tuples of distinct vertices, and let $\mathrm{IP}(k)$ and $\mathrm{RW}(k)$ denote, respectively, the $k$-particle inter…

Tóth's conjecture. For $d\ge 3$, infinite orbits arise in the interchange process on $\mathbb{Z}^d$ run for a sufficiently long time, whereas for $d\in\{1,2\}$ no infinite orbits a…

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Let $k\geq 2$, let $\mathcal S_k$ be the symmetric group on $k$ letters, and let $g:\mathcal S_k\to\mathbb R$ be any function. For nonnegative numbers…

Let $X_N=\{0,1\}^N$ be the vertex set of the $N$-dimensional hypercube, let $X_{N,2}=\{\{i,j\}:1\leq i<j\leq N\}$ be the set of coordinate pairs, and let $q:X_{N,2}\to[0,\infty)$ b…