Extend metastability results to the critical regime

Prove that the two-species zero-range process with attraction exponent \(\alpha=1\), general irreducible underlying random walks, and multiple particle species exhibits the same condensation and dynamical metastability results as in the supercritical regime, in particular establishing the local metastability condition for each candidate metastable set \(\mathcal{E}^{x}_{N}\), and thereby obtaining the corresponding metastable transition behavior and negligibility of the remainder set.

Background

The paper proves dynamical metastability for the two-species zero-range process only in the supercritical regime α>1\alpha>1. In the critical regime α=1\alpha=1, the stationary condensation result is established, and the authors state that the remaining dynamical conclusions should also hold. The main unresolved component is the local metastability, or attractor, condition: the candidate metastable valleys ENx\mathcal{E}^{x}_{N} are substantially larger at criticality, so the capacity estimate used for α>1\alpha>1 does not suffice to prove rapid local mixing.

For the single-species model with symmetric random walks, a delicate superharmonic test function resolves the analogous critical difficulty. The authors indicate that this strategy is not expected to extend directly to general non-reversible, multi-species settings, so new methods are needed to establish the critical-case theorem in the setting considered here.

References

We strongly believe that the same results hold in the remaining critical case $\alpha=1$. Indeed, the condensation result is already established in Theorem \ref{thm:cond}, and parts (2) and (3) of Theorem \ref{thm:main} can be checked in a similar manner, albeit with significantly more technical details. The main bottleneck in this generalization lies in part (1) of the main theorem, namely the local metastability condition. This presents a substantially harder challenge because, when $\alpha=1$, each (potentially) metastable set $\mathcal{E}{x}_{N}$, $x\in S_{\star}$, becomes significantly larger, so that a crude capacity estimate (as performed in Proposition \ref{prop3} for the supercritical case) no longer suffices to ensure rapid local mixing. In , this difficulty was overcome for the single-species zero-range process in the symmetric random walk case through the construction of a delicate superharmonic test function on the annulus region of $\mathcal{E}{x}_{N}$, combined with same the capacity estimate localized near the central configuration $\bm{\xi}{x}_{N}$. We do not expect this strategy to extend, at least directly, to general non-reversible, multi-species settings, suggesting that entirely new ideas will be required to resolve the critical case.

— Condensation and metastability in the supercritical two-species zero-range process via resolvent and $H^1$-approximation  (2609.03821 - Kim et al., 3 Sep 2026) in Remark immediately following Theorem 1.2 (after Remark 1.4, before Section 1.4)