Convergence to equilibrium for strongly reinforced Pólya urns

Let a WARM with parameter $\alpha>1$ have empirical proportion vector $\vec{X}_t$. An equilibrium is a fixed point of the associated mean-field dynamical system; it is linearly-stable if all eigenvalues of its Jacobian have negative real parts, and critical otherwise.

Convergence to equilibrium. For any WARM with $\alpha>1$, there exists a random vector $\vec{X}=(X_1,\dots,X_n)$, supported on the set of linearly-stable and critical equilibria, such that

$$\mathbb{P}(\vec{X}_t\rightarrow \vec{X})=1.$$

The preceding theorem establishes structural information about accumulation points and positive-probability convergence to each linearly-stable equilibrium under additional hypotheses, while this conjecture predicts almost-sure convergence for every WARM with $\alpha>1$; the general case remains open.