8 problems

Pemantle's conjecture. CNC implies CNA. In particular, the equivalent formulation considered here is that $R^+$ implies CNA.

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

Let $C_N$ be a cycle, let $K\ge3$, and consider the $K$-type competing urn scheme with a nonzero initial configuration. For each type $j$, let balls of type $j$ have independent Po…

Let $C_k$ be the cycle graph of length $k$, and consider the $s$-type urn model on $C_k$, with balls of $s$ colours and each urn monochromatic. At each step, a uniformly chosen bal…

Let $G$ be a finite connected graph, and let $x(n)$ denote the associated urn process. Convergence to a point or interval conjecture. There exists either a point $v(G)$ such that…

Strong reinforcement conjecture. If $s_{\infty}<\infty$, then

Partitioned-urn conjecture. The variables $xi_{ij}$ are negatively associated conditional on

Let $(A(t),B(t))$ be the fast continuous-time embedding of the simple harmonic urn, let … and write $\mathbb{E}_n$ for expectation conditional on $A(0)=n$ and $B(0)=0$. Let…