9 problems

Oliveira's conjecture. The mixing time of the interchange process on $G$ should be of the same order as the mixing time of the product chain.

Let $\Gamma$ be a hypergraph with non-negative weights. For every $k\in\mathbb{Z}_{\geq 1}$, let $\omega_k=\omega_k(\Gamma)$ denote the smallest new eigenvalue of the Laplacian…

Let $\Gamma=([n],w)$ be an arbitrary hypergraph with non-negative weights. Let $\mathcal{KMP}_k$ be the discrete KMP process with $k$ indistinguishable particles on $\Gamma$, and l…

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

The generalized Aldous conjecture. For any choice of weights $c$, the spectral gap of the generalized interchange process coincides with the spectral gap of its single-particle pro…

Let $V$ be a set of $n$ vertices, let ${\mathcal S}_n$ be the symmetric group of permutations of $V$, and let $\a=\{\a_A:A\subset V\}$ be a collection of nonnegative weights defini…

Given the quantum Heisenberg ferromagnet on the Hamming graph, let $\beta>0$ denote the inverse temperature. The results establish no magnetic ordering for $\beta<1/4$ and magnetic…

Let $m(\beta)$ denote the spontaneous magnetization of the quantum Heisenberg ferromagnet, represented through the interchange process on $[-r,r]^3$ at inverse temperature $\beta$.…

Let $\cG$ be a finite connected simple graph with vertex set of size $n$. Write $\Cay(\cG)=\Cay(S_n,E(\cG))$ for the Cayley graph of the symmetric group generated by the transposit…