The b-conjecture and Airyβ soft-edge limit for the Jack–Plancherel process

For every $\beta>0$ , the standard edge-scaled Jack–Plancherel process converges at the soft edge, in its multi-time finite-dimensional statistics (in particular, in moments), to the corresponding Airy $_\beta$ process. Equivalently, for every finite collection of times $t_1,\ldots,t_k$ and every admissible polynomial test function $P$ , one has $\lim_{n\to\infty}\mathbb{E}\!\left[P\!\left(X_n^{(\beta)}(t_1),\ldots,X_n^{(\beta)}(t_k)\right)\right]=\mathbb{E}\!\left[P\!\left(\mathcal{A}_\beta(t_1),\ldots,\mathcal{A}_\beta(t_k)\right)\right]$ , where $X_n^{(\beta)}$ is the edge-scaled Jack–Plancherel process and $\mathcal{A}_\beta$ is the Airy $_\beta$ process.

References

Primary source

arXiv

Additional references

Airy limit for the Jack process and topological expansion — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

A September 2026 paper proves major limit theorems for this random-partition model, but does not claim to settle every version of the conjecture.

The problem concerns whether the Jack–Plancherel process has an Airy soft-edge limit and yields the predicted expansion associated with the $b$ -conjecture. The latest paper links this discrete model to Airy processes and develops the relevant asymptotic formulas.

Known results

Féray and Dołęga proved the law of large numbers and central limit theorem for Jack–Plancherel measure.
Féray and Dołęga conjectured the edge scaling limit.
Guionnet and Huang later proved that the edge limit is governed by the $\beta$ -Tracy–Widom law.

September 2026 progress

The paper establishes multi-time moment convergence, identifies the equal-time limit with Airy $\beta$ statistics for $\beta \ge 1$ , and derives an asymptotic $\beta$ -topological expansion, including a nonnegative $\beta=2$ formula for Witten–Kontsevich intersection numbers. This is substantial progress, but the abstract does not claim a complete resolution of every formulation of the $b$ -conjecture; the claims remain unverified here.

Current status (as of September 2026): Major soft-edge and expansion results are claimed, while a complete resolution of the $b$ -conjecture remains open or unestablished.

Sources

No solutions have been posted yet.