9 problems

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

Let $\sigma\in {\mathcal M}_1((0,1/2])$ and let $\beta_m,\beta_s\in(0,1]$. Let $K_{\sigma,\beta_m,\beta_s}$ denote the Markov kernel governing the coagulation-fragmentation process…

Let $f=\sum\varepsilon_i x^i$ be a random polynomial with independent coefficients in $\mathbb F_p$. Consider the degrees of all irreducible factors of $f$, normalized by the total…

Let $f=\sum\varepsilon_i x^i$ be a random polynomial with independent coefficients in $\mathbb F_p$. Write $D(f)$ for the maximal degree of an irreducible factor of $f$ and…

Poisson–Dirichlet correlation conjecture. One has

Macroscopic-loop merge-rate conjecture. The number of endpoint pairs that allow two macroscopic loops $\gamma$ and $\gamma'$ to merge is approximately

For each $n=1,2,\dots$, let $N(n)$ be the random integer whose prime-factor proportions are denoted by $P_i(N(n))$, and let $(L_i)_{i\geq 1}$ have the Poisson--Dirichlet distributi…

Aldous's conjecture. This random partition has Poisson-Dirichlet$(1)$ distribution.

Let $\rho\trimmed$ be a stationary random measure on $\mathbb{R}$ with intensity-one asymptotic density, let $\mathscr{K}$ be the disjoint union of…