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We consider the symmetric inclusion process on a general finite graph. Ourmain result establishes universal upper and lower bounds for the spectral gapof this interacting particle system in terms of the spectral gap of the randomwalk on the same graph. In the regime in which the gamma-like reversiblemeasures of the particle systems are log-concave, our bounds match, yielding aversion for the symmetric inclusion process of the celebrated Aldous' spectralgap conjecture originally formulated for the interchange process. Finally, bymeans of duality techniques, we draw analogous conclusions for an interactingdiffusion-like unbounded conservative spin system known as Brownian energyprocess.

Spectral gap of the symmetric inclusion process