8 problems

Let $\Sigma_1$ be the simplex of monotone non-negative sequences $x=(x_1,x_2,\ldots)$ with unit sum. The split-merge Markov operator $T$ acts on $\Sigma_1$ by sampling two elements…

Let $\Lambda$ be a finite region of the lattice, let $u\in[0,1]$ and let $\beta$ be the inverse temperature. Consider the loop model associated with the spin-nematic system, and le…

Poisson–Dirichlet convergence conjecture. The rescaled cycle sizes converge in distribution to

Let $\tt FV(\alpha,\theta)$ be the Fleming–Viot diffusion with parameters $\alpha\in(0,1)$ and $\theta>-\alpha$, and let $\tt PDRM(\alpha,\theta)$ denote its stationary Poisson–Dir…

Let a Fleming–Viot diffusion be given, and consider its projection onto the Kingman simplex. Feng–Sun projection conjecture. Petrov's two-parameter Poisson–Dirichlet diffusions sho…

Let $L^{(1)},L^{(2)},\dots$ denote loop lengths in decreasing order, and normalize the macroscopic loop lengths by their total length. A random partition of $[0,1]$ is obtained fro…

Under the preceding critical-temperature conjecture, let $\xi\in\Delta_1$ be the weak limit of the ordered normalized cycle-length vector. Poisson–Dirichlet law conjecture. The dis…

Let $d\geq 3$ and consider the cycle or loop model on the cubic lattice $\Lambda_n=\{1,\dots,n\}^d$ in $\mathbb{Z}^d$. The quantities $\eta_{\infty}(\beta)$ and…