For every finite connected weighted graph G=(V,E,w), the spectral gap of the interchange process on G equals the spectral gap of the continuous-time random walk on G: lambda_1^IP(G)=lambda_1^RW(G), where lambda_1^IP(G) and lambda_1^RW(G) denote the respective first nonzero eigenvalues of the generators.

Aldous's spectral gap conjecture

Aldous formulated the conjecture around 1992: for every weighted graph, the interchange process and random walk have equal spectral gaps, $\lambda_1^{\mathrm{IP}}(G)=\lambda_1^{\mathrm{RW}}(G)$ . Caputo, Liggett, and Richthammer proved it for arbitrary weighted graphs.

Known results

Caputo, Liggett, and Richthammer, 2009 (published 2010): proved the conjecture for every finite weighted graph, using the octopus inequality.

September 2026 Plücker-positivity development

Submitted September 3, 2026, Haoran Zhu’s preprint claims a positivity theorem for universal Plücker coordinates and connects positivity of a normalized $(2,2)$ coordinate at a critical specialization with the octopus inequality. It presents related hypergraph inequalities but does not claim a new proof of Aldous’s conjecture; the new claims are unverified.

Current status (as of September 2026): The original conjecture is settled by Caputo, Liggett, and Richthammer; the new strengthening of its octopus-inequality machinery remains unverified.

Aldous's spectral gap conjecture

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Additional references

Progress summary

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